A blow-up approach for singular elliptic problems with natural growth in the gradient
aa r X i v : . [ m a t h . A P ] F e b A BLOW-UP APPROACH FOR SINGULAR ELLIPTIC PROBLEMS WITHNATURAL GROWTH IN THE GRADIENT
SALVADOR LÓPEZ-MARTÍNEZInria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000Lille, France
Email adress: [email protected]
Abstract.
We prove existence and nonexistence results concerning elliptic problems whosebasic model is − ∆ u + µ ( x ) |∇ u | ( u + δ ) γ = λu p , x ∈ Ω ,u > , x ∈ Ω ,u = 0 , x ∈ ∂ Ω , where Ω ⊂ R N ( N ≥ is a bounded smooth domain, λ > , p > , δ ≥ , γ > and µ ∈ L ∞ (Ω) . The main achievement resides in handling a possibly singular ( δ = 0 ) first orderterm having a nonconstant coefficient µ in the presence of a superlinear zero order term. Ourapproach for the existence results is based on fixed point theory. With the aim of applyingit, a previous analysis on a related non-homogeneous problem is carried out. The required apriori estimates are proven via a blow-up method. Introduction
Let Ω ⊂ R N ( N ≥ be a bounded domain of class C , g : Ω × (0 , + ∞ ) → R be a Carathéodoryfunction, and f : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function. In this work we will study theexistence of solution to elliptic problems of the following form:( P λ ) − ∆ u + g ( x, u ) |∇ u | = λf ( u ) , x ∈ Ω ,u > , x ∈ Ω ,u = 0 , x ∈ ∂ Ω , where λ > is a parameter. The precise conditions on functions g, f and the statements ofthe main results regarding problem ( P λ ) will be shown in Section 2. For the sake of a clearpresentation, we consider for now a model problem:(1.1) − ∆ u + µ ( x ) |∇ u | ( u + δ ) γ = λu p , x ∈ Ω ,u > , x ∈ Ω ,u = 0 , x ∈ ∂ Ω , Key words and phrases.
Nonlinear elliptic equations, Singular gradient terms, Blow-up argument.
MSC: where δ ≥ , γ > , p > and µ ∈ L ∞ (Ω) . Our main goal is to allow µ to be nonconstant andeven sign-changing, paying special attention to the singular case δ = 0 .Problem (1.1) for µ ≡ becomes semilinear; as p > , it is usually referred to as superlinear.This superlinear case is classical and has been extensively studied in the literature. Indeed, bothvariational (in [1]) and topological (in [21, 25]) methods can be used to prove the existence ofa solution to (1.1) for all λ > provided µ ≡ and p ∈ (1 , ∗ − , where ∗ = NN − . It iswell-known that Pohozaev’s identity (see [36]) implies that the restriction p < ∗ − is necessaryfor the existence of solution to the superlinear problem if the domain is starshaped.The study of problem (1.1) for a nontrivial µ was initiated in [35]. There, the authors consider p ∈ (1 , ∗ − , δ > and µ ≡ constant > . In this setting, they prove several results whichcan be divided into two classes: on the one hand, those which lead to a set of solutions similarto that for the classical semilinear problem (i.e. existence for all λ > ) and, on the otherhand, those which present differences such as nonexistence of solution for λ > small. In fact,the semilinear-like behavior is achieved provided γ > , while the differences appear if either γ ∈ (0 , or γ = 1 and µ > p . In several proofs, the authors of [35] make use of the change ofvariable(1.2) v = ψ ( u ) = Z u e − R s µ ( t + δ ) γ dt ds. It is easy to check formally that, if µ ≡ constant and δ > , then the transformation (1.2) turns(1.1) into a semilinear problem in the variable v . Thus, roughly speaking, the gradient termis removed and semilinear techniques (such as variational methods) can be applied in general.However, such a transformation can be performed only if µ is constant.We remark that problem (1.1) in the case γ = 1 is specially interesting because of the condition µ > p that appears in [35], which shows that the interaction between the gradient term and thesuperlinear term in (1.1) plays an important role. To this respect, some results concerning thecase γ = 1 (always with δ > and µ ≡ constant) that improve those in [35] in some directions canbe found in [3, 29]. In the first work, the authors prove nonexistence of solution for every λ > small enough provided µ ≥ p , while in the second one the authors prove existence of solution forevery λ > provided µ < ∗ − − p ∗ − . We point out that, in both mentioned works, no restrictionon p from above is imposed. Nevertheless, if p ≥ ∗ − , then the condition µ < ∗ − − p ∗ − requiredby the existence result in [29] forces µ to be negative. We also stress that a blow-up argumentis employed in [29] in order to obtain a priori estimates, even though the change of unknown(1.2) is strongly used in order to get rid of a quadratic gradient term from the general problemthat the authors consider and, in consequence, non-constant functions µ cannot be handled withtheir approach.Still focusing on problem (1.1) with γ = 1 , δ > and µ ≡ constant, the range ∗ − − p ∗ − ≤ µ < p has not been considered in the literature to our knowledge. However, in this particular situationit is easy to see that the transformation (1.2) turns (1.1) into a semilinear equation whosenonlinearity presents supercritical growth at infinity and subcritical growth at zero. Therefore, [7,Theorem 8] implies (after undoing the change of unknown) that there exists at least a solutionto (1.1) for every λ > large enough. Again, last (immediate) result is based on the change ofunknown, so does not cover problem (1.1) with nonconstant µ .On the other hand, the singular case δ = 0 has been dealt with recently in [18], one more timefor µ ≡ constant > (see also [14,19] for similar singular problems that involve a nonzero sourceterm). The authors of [18] show that, if γ ∈ (0 , , then the situation is similar to the nonsigular LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 3 case δ > . Indeed, they prove a nonexistence result for λ > small and an existence result for λ > large. On the contrary, for γ ≥ they prove nonexistence results for all λ > . This factexposes the remarkable influence of a strong singularity in the equation. As far as we know, the µ constant case for δ = 0 has not been studied in the literature.To sum up, in the present work we aim to develop an approach that permits to deal with x − dependent µ in problem (1.1) and also with singular lower order terms, i.e. δ = 0 . In orderto do so, we will employ topological methods. More precisely, we will find solutions to (1.1) asfixed points of certain compact operator that will be defined in Section 4. The well-definition ofsuch an operator will require the well-posedness of the following problem:(1.3) − ∆ u + µ ( x ) |∇ u | u = h ( x ) , x ∈ Ω ,u > , x ∈ Ω ,u = 0 , x ∈ ∂ Ω , where (cid:12) h ∈ L q (Ω) for some q > N .Singular problems of this kind have risen interest in the recent years. In fact, the existence ofsolution with k µ k L ∞ (Ω) < has been proven in [11], and extended to k µ k L ∞ (Ω) < in [32]. As faras the uniqueness of solution is concerned, some results are known for problems similar to (1.3),even though they require either the singularity to be milder or µ to be constant (see [4, 8, 16]).We will prove that uniqueness for problem (1.3) holds provided k µ k L ∞ (Ω) < ; the proof is basedon a comparison principle that we state in Section 2 and prove in Section 4. Furthermore, wewill show that the condition k µ k L ∞ (Ω) < is natural by proving a nonexistence result provided µ > in a neighborhood of ∂ Ω ; the proof follows the ideas in [18, Lemma 2.5]. In next statementwe summarize the new uniqueness and nonexistence results that we prove about problem (1.3)and we include the previously known ( [11, 32]) existence part for completeness. Theorem 1.1.
Let (cid:12) h ∈ L q (Ω) for some q > N and let (cid:12) µ ∈ L ∞ (Ω) . The followingstatements hold true:(1) If k µ k L ∞ (Ω) < , then there exists a unique finite energy solution to problem (1.3) .(2) If there exist an open domain ω ⊂⊂ Ω and a constant τ > such that µ ( x ) ≥ τ and h ( x ) = 0 , both for a.e. x ∈ Ω \ ω, then problem (1.3) admits no solution. We point out that existence and uniqueness results for problem (1.3) are known for generalnonnegative µ ∈ L ∞ (Ω) (i.e. without assuming that k µ k L ∞ (Ω) < ) provided h is locally boundedaway from zero (see [2] for the existence and [16] for the uniqueness). Thus, we clarify thatthe condition h ≡ near the boundary in item (2) of Theorem 1.1 is also natural for havingnonexistence of solution.Once we have shown that problem (1.3) is well-posed, we will be able to define a compactoperator whose fixed points are solutions to ( P λ ) (see Section 4). A version of a result in [27](see [21]) will assure the existence of a fixed point of the operator.As it is mandatory for fixed point theorems, we will prove the existence of a priori estimateson the solutions to a problem related to ( P λ ). To this task, we will adapt the blow-up methoddue to [25]. Roughly speaking, this technique consists of assuming by contradiction that thereexists a sequence of solutions whose norms blow up as n tends to infinity. The conclusionfollows by passing to the limit in a problem satisfied by a certain normalized sequence. In fact,the limit function is a solution to a problem which, however, does not admit any solution by BLOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH virtue of some Liouville type result. Therefore, one gets a contradiction, so any sequence ofsolutions must be bounded. In this context, the difficulties that we find are twofold. Firstly,the normalized sequence, say { v n } , satisfies an equation having a lower order term of type |∇ v n | v n + δ n , where ≤ δ n → as n → + ∞ . If we aim to pass to the limit, then we need tofind positive lower bounds on { v n } , otherwise the lower order term may blow up as n → + ∞ .And lastly, we arrive to a limit problem, having a quadratic gradient lower order term of theform |∇ v | v , for which nonexistence Liouville type results are not known in the literature (a non-exhaustive list of references for Liouville type results about problems depending on the gradientis [23, 24, 30, 34, 37–39]).We overcome the first of the difficulties by proving Hölder estimates in spite of the singularquadratic term. The proof follows the ideas of [28], which have been widely used for singularproblems (see [2,6,15,17,22,31,32], among others). We will show that these estimates yield in turnpositive lower estimates from below and this will be enough to pass to the limit. Regarding thesecond difficulty, we observe that the limit equation does admit a convenient change of unknownwhich gets rid of the gradient term, so that we may apply classical Liouville type results (seeSection 3 below).We state here the main existence result for problem (1.1) in the case γ = 1 . Theorem 1.2.
Let p > , γ = 1 , δ ≥ and µ ∈ L ∞ (Ω) . The following statements hold true:(1) If µ ∈ C (Ω) and there exist two real numbers τ, σ such that σ − < τ ≤ σ < ∗ − − p ∗ − and τ ≤ µ ( x ) ≤ σ for all x ∈ Ω , then there exists at least a solution to (1.1) for every λ > . If, in addition, either µ ≥ or δ > , then there exists at least a finite energysolution to (1.1) for every λ > .(2) If δ = 0 and there exist an open domain ω ⊂⊂ Ω and a constant τ > such that µ ( x ) ≥ τ for a.e. x ∈ Ω \ ω , then problem (1.1) admits no solution for any λ > . Note that, in the first item of Theorem 1.2, µ is allowed to change sign unless p ≥ ∗ − , inwhich case µ is necessarily negative. We also point out that the smallness condition σ < ∗ − − p ∗ − in Theorem 1.2 is natural since, in fact, problem (1.1) has no bounded solutions provided γ = 1 , δ = 0 , µ ≡ constant ∈ h ∗ − − p ∗ − , (cid:17) and Ω is starshaped (see Remark 2.5 below). Moreover,we will show later that, strengthening the smallness condition conveniently (in terms of p, N ),one may assume µ to be either continuous only in a neighborhood of ∂ Ω , or merely boundedin Ω . Also about the existence part of the theorem, it is worth to point out that we need tocontrol µ from below in order to prove the Hölder estimates that we mentioned. However, if µ is constant, i.e. σ = τ , then the condition σ − < τ becomes σ = τ < , which means norestriction since σ < ∗ − − p ∗ − < . On the other hand, we stress that the case µ ≡ and δ = 0 remains unsolved, i.e. there are neither existence nor nonexistence results about problem (1.1)for γ = 1 , δ = 0 , µ ≡ , p > in the literature.Next result shows that our approach allows also to prove existence for γ > and for all λ > . Theorem 1.3.
Let p > , γ > , δ ≥ and µ ∈ L ∞ (Ω) . The following statements hold true:(1) If δ > and µ satisfies (1.4) k µ + k L ∞ (Ω) + k µ − k L ∞ (Ω) < δ γ − , then there exists at least a finite energy solution to (1.1) for every λ > .(2) If δ = 0 and there exist an open domain ω ⊂⊂ Ω and a constant τ > such that µ ( x ) ≥ τ for a.e. x ∈ Ω \ ω , then problem (1.1) admits no solution for any λ > . LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 5
Notice that, if µ ≡ constant > , then condition (1.4) becomes µ < δ γ − . It is known that sucha smallness condition is not needed if µ is a positive constant (see [35, Theorem 1.2]). Therefore,we presume that the assumption (1.4) is technical, even though we cannot avoid it since it isused to prove the Hölder estimates mentioned above. Concerning the nonexistence statement forthe singular case δ = 0 , it is proven again following closely [18, Lemma 2.5]. We stress that thefact that the singularity is strong, namely γ > , allows to take µ > near ∂ Ω (in contrast tothe case γ = 1 , for which µ > near the boundary was needed).We will be able to go beyond Theorem 1.2 and Theorem 1.3 and deal with any γ > . Indeed,for γ ∈ (0 , and δ ≥ , we will show that existence of solution holds for every λ > largeenough. We will also deal with γ ≥ , δ > and µ a bounded function with arbitrary size, i.e.we remove the size restrictions on µ in Theorem 1.2 and Theorem 1.3 as long as δ > , eventhough λ must be large. The statement of the result is the following. Theorem 1.4.
Let p > , γ > , δ ≥ and µ ∈ L ∞ (Ω) . If γ ≥ , assume in addition that δ > . Assume also one of the following two conditions:(1) p < N +1 N − ,(2) p < ∗ − and ≤ µ ∈ C (Ω) .Then, there exists λ > such that there exists at least a solution u λ to (1.1) for every λ > λ satisfying lim λ → + ∞ k u λ k L ∞ (Ω) = 0 . Moreover, if either µ ≥ or δ > , then u λ is a finite energysolution. If γ ∈ (0 , , last result is consistent with the known results for µ ≡ constant > whichassure nonexistence for λ > small (see [35] for δ > and [18] for δ = 0 ). On the contrary, aswe pointed out above, for γ > we would expect an existence result for every λ > (even for λ small) without size restrictions on µ . As far as the case γ = 1 and δ > is concerned, weignore whether an existence result for λ > small and general µ should be expected or not. Twoexceptions are the ranges µ < ∗ − − p ∗ − and µ ≥ p , for which existence (see Theorem 1.2 aboveand [29]) and nonexistence (see [4, 35]) for λ > small are known respectively. In other words,the existence of solution to (1.1) for γ = 1 , δ > , ∗ − − p ∗ − ≤ µ < p and λ > small remains asan open problem, even for µ ≡ constant > .We organize the paper as follows. In Section 2 we introduce the conditions on f, g and thestatements of the general results about problem ( P λ ). Section 3 is devoted to proving someLiouville type results as well as several propositions that provide the estimates via the blow-upmethod. In Section 4 we prove the results that we state in Section 2 and in the Introduction.Finally, in the Appendix we gather some technical results that are required throughout the paper. Acknowledgments.
The problems considered in this work have been proposed by T. Leonori.The research was initiated with his collaboration during one week in Granada and another weekin Rome, when most of the ideas contained here emerged. Moreover, for the conclusion of thepaper, the interesting ideas, comments and corrections by J. Carmona have also supposed a morethan remarkable contribution. In any case, their supervision and support have been essentialduring the research period. This is why the author wants to warmly thank both, collaboratorsand friends, for being examples of altruism to follow in this competitive and sometimes hostileworld of research. The author wants to thank also D. Ruiz and C. De Coster for their kind anduseful suggestions and corrections.
BLOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH Hypotheses and main results
Let us consider a continuous function f : [0 , + ∞ ) → [0 , + ∞ ) and a Carathéodory function g : Ω × (0 , + ∞ ) → R . We will assume throughout this paper that g satisfies the followingcondition:(2.1) ∃ g ∈ C ((0 , + ∞ )) : | g ( x, s ) | ≤ g ( s ) a.e. x ∈ Ω , ∀ s > . Last hypothesis is essentially the minimal condition that g must satisfy so that the weak formu-lation of ( P λ ) is well defined: Definition 2.1.
Let f : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function and g : Ω × (0 , + ∞ ) → R be a Carathéodory function satisfying (2.1). A solution to ( P λ ) is a function u ∈ H loc (Ω) ∩ L ∞ (Ω) such that ∀ ω ⊂⊂ Ω , ∃ c > u ( x ) ≥ c a.e. x ∈ ω, ∃ γ > u γ ∈ H (Ω) , and Z Ω ∇ u ∇ φ + Z Ω g ( x, u ) |∇ u | φ = Z Ω f ( u ) φ ∀ φ ∈ C c (Ω) . Besides, we will say that u is a finite energy solution to ( P λ ) if it is a solution to ( P λ ) with γ = 1 ,i.e. if u ∈ H (Ω) . Remark 2.2.
It can be proven by following [15, Appendix] that, in the previous definition, onecan take test functions φ ∈ H (Ω) ∩ L ∞ (Ω) with compact support. Moreover, if u is a finiteenergy solution, then one can take any test function belonging to H (Ω) ∩ L ∞ (Ω) .Let us fix σ ∈ R . The following condition, stronger than (2.1), means a more precise controlon g from above:( g ) ∃ g ∈ C ((0 , + ∞ )) : − g ( s ) ≤ g ( x, s ) ≤ σs a.e. x ∈ Ω , ∀ s > . Let us consider also the monotonicity condition:( g ր ) s sg ( x, s ) is nondecreasing for a.e. x ∈ Ω . We state next a comparison principle that will be the key for proving the uniqueness partof Theorem 1.1. The proof follows essentially the arguments in [15], which in turn are inspiredby [5].
Theorem 2.3.
Let (cid:12) h ∈ L loc (Ω) and let g : Ω × (0 , + ∞ ) → R be a Carathéodory functionsatisfying ( g ր ) and ( g ) for some σ ∈ (0 , . Let u, v ∈ C (Ω) ∩ W ,N loc (Ω) , with u, v > in Ω , besuch that Z Ω ∇ u ∇ φ + Z Ω g ( x, u ) |∇ u | φ ≤ Z Ω h ( x ) φ and (2.2) Z Ω ∇ v ∇ φ + Z Ω g ( x, v ) |∇ v | φ ≥ Z Ω h ( x ) φ (2.3) for every ≤ φ ∈ H (Ω) ∩ L ∞ (Ω) with compact support. Suppose also that the following boundarycondition holds: (2.4) lim sup x → x ( u ( x ) − σ − v ( x ) − σ ) ≤ ∀ x ∈ ∂ Ω . Then, u ≤ v in Ω . LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 7
We will present below four existence theorems that represent the core of this work. The proofsare based on a fixed point result. The required a priori estimates are obtained via a blow-upmethod (see Section 3).Let us fix p > and δ ≥ . We establish now a growth condition and two limit conditions on f which make it behave like a superlinear power:( f ∗ ) ∃ a ≥ s p ≤ f ( s ) ≤ a ( s + δ ) p ∀ s ≥ . ( f ∞ ) ∃ L ∈ (0 , + ∞ ) : lim s → + ∞ f ( s ) s p = L. ( f ) lim s → f ( s ) s = 0 . Observe that ( f ) implies in particular that f (0) = 0 .On the other hand, for fixed δ ≥ and σ, τ ∈ R , we set the following growth restriction on g which will allow us to prove certain Hölder estimates (see the Appendix below):( g ∗ ) ( σ − < τ ≤ σ < ,τ ≤ ( s + δ ) g ( x, s ) ≤ σ a.e. x ∈ Ω , ∀ s > . We will also need the following limit condition on g at infinity for p > :( g ∞ ) ( ∃ µ ∈ C (Ω) : max x ∈ Ω µ ( x ) < ∗ − − p ∗ − , lim s → + ∞ k sg ( · , s ) − µ k L ∞ (Ω) = 0 . A simple limit condition on g at zero will be required too:( g ) ∃ lim s → ( s + δ ) g ( x, s ) a.e. x ∈ Ω . Observe that, if δ > , then ( g ) is equivalent to g ( x, · ) being continuous at s = 0 for a.e. x ∈ Ω .We are ready to state our first general existence result: Theorem 2.4.
Let f : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function and g : Ω × (0 , + ∞ ) → R be a Carathéodory function. For δ ≥ , σ, τ ∈ R and p > , assume that f satisfies ( f ∗ ) , ( f ∞ ) and ( f ) , and that g satisfies ( g ∗ ) , ( g ∞ ) and ( g ) . Then, there exists at least a solution to ( P λ ) for every λ > . If, in addition, either g ≥ or g ( x, · ) is continuous at s = 0 a.e. x ∈ Ω , thenthere exists at least a finite energy solution to ( P λ ) for every λ > . Remark 2.5.
We point out that the smallness condition max x ∈ Ω µ ( x ) < ∗ − − p ∗ − in Theorem 2.4coming from condition ( g ∞ ) is necessary for the existence of solutions to ( P λ ), at least in themodel case of Ω a satarshaped domain, f ( s ) = λs p for some λ > , p ∈ (1 , ∗ − , and sg ( x, s ) ≡ µ for some constant µ < . Indeed, assume by contradiction that µ ∈ h ∗ − − p ∗ − , (cid:17) and that < u ∈ H (Ω) ∩ L ∞ (Ω) satisfies − ∆ u + µ |∇ u | u = λu p in Ω . Then, v = cu − µ satisfies − ∆ v = v p − µ − µ in Ω for some c > . Therefore, since p − µ − µ ≥ ∗ − and Ω is starshaped, thewell-known Pohozaev’s identity (see [36]) yields a contradiction. BLOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH
Let us fix again p > . Next hypothesis is a limit condition on g at infinity weaker than ( g ∞ )as it is only required to hold in a neighborhood of ∂ Ω .( f g ∞ ) ( ∃ ω ⊂⊂ Ω , µ ∈ C (Ω \ ω ) : max x ∈ Ω \ ω µ ( x ) < ∗ − − p ∗ − , lim s → + ∞ k sg ( · , s ) − µ k L ∞ (Ω \ ω ) = 0 . The following result shows that, assuming a stronger control on g from above, one can relaxthe limit condition on g at infinity to obtain a solution to ( P λ ). Theorem 2.6.
Let f : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function and g : Ω × (0 , + ∞ ) → R bea Carathéodory function. For δ ≥ , σ, τ ∈ R and p > , assume that f satisfies ( f ∗ ) , ( f ∞ ) and ( f ) , and that g satisfies ( g ∗ ) , ( f g ∞ ) and ( g ) . Assume in addition that σ ≤ N − ( N − p . Then,there exists at least a solution to ( P λ ) for every λ > . If, in addition, either g ≥ or g ( x, · ) is continuous at s = 0 a.e. x ∈ Ω , then there exists at least a finite energy solution to ( P λ ) forevery λ > . Next theorem shows that the limit conditions at infinity are actually not essential to obtainsolutions to ( P λ ). In return, one has to assume an even stronger control on g from above. Theorem 2.7.
Let f : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function and g : Ω × (0 , + ∞ ) → R bea Carathéodory function. For δ ≥ , σ, τ ∈ R and p > , assume that f satisfies ( f ∗ ) and ( f ) ,and that g satisfies ( g ∗ ) and ( g ) . Assume in addition that σ ≤ N +1 − ( N − p . Then, there existsat least a solution to ( P λ ) for every λ > . Moreover, the following holds.(1) If either g ≥ or g ( x, · ) is continuous at s = 0 a.e. x ∈ Ω , then there exists at least afinite energy solution to ( P λ ) for every λ > .(2) If δ = 0 , then there exists C > such that, for every λ > and for every solution u to ( P λ ) , the estimate λ p − k u k L ∞ (Ω) ≤ C holds. Let s ∈ (0 , , σ, τ ∈ R and p > . Consider the following growth conditions near zero:( e f ∗ ) ∃ a ≥ s p ≤ f ( s ) ≤ as p ∀ s ∈ (0 , s ) . ( e g ∗ ) ( σ − < τ ≤ σ < ,τ ≤ sg ( x, s ) ≤ σ a.e. x ∈ Ω , ∀ s ∈ (0 , s ) . Notice that condition ( e f ∗ ) implies that lim s → f ( s ) s = 0 and, in particular, f (0) = 0 . On theother hand, we point out that, if τ > , then condition ( e g ∗ ) forces g to be singular at s = 0 .However, if ( e g ∗ ) holds for some τ < and σ > , then g may be rather general at s = 0 since itcan be continuous as well as unbounded from above and from below.Our next existence theorem will require neither growth nor limit conditions at infinity on f and g . In exchange, the existence of solution holds only for every λ > large enough. Theorem 2.8.
Let f : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function and g : Ω × (0 , + ∞ ) → R be a Carathéodory function. For s ∈ (0 , , σ, τ ∈ R and p > , assume that f satisfies ( e f ∗ ) andthat g satisfies ( e g ∗ ) . Assume in addition that there exists lim s → sg ( x, s ) for a.e. x ∈ Ω , andalso that σ ≤ N +1 − ( N − p . Then, there exists λ > such that there exists at least a solution u λ to ( P λ ) for every λ > λ satisfying lim λ → + ∞ k u λ k L ∞ (Ω) = 0 . If, in addition, either g ≥ or g ( x, · ) is continuous at s = 0 a.e. x ∈ Ω , then u λ is a finite energy solution. LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 9
Regarding Theorem 2.8, observe that, if p > N +1 N − , then σ < , so g ( x, s ) < for s near zero.Thus, for instance, the particular case g ( x, s ) = µs γ and f ( s ) = s p is not covered by Theorem 2.8if γ ∈ (0 , , p ∈ h N +1 N − , ∗ − (cid:17) and µ > is constant. The result that comes next does cover thisparticular case. In fact, for p ∈ (1 , ∗ − , next theorem allows to consider general continuousfunctions g satisfying(G) ∃ G ∈ L ((0 , s )) : 0 ≤ g ( x, s ) ≤ G ( s ) ∀ ( x, s ) ∈ Ω × (0 , s ) . ( e g ) ( ∃ µ ∈ C (Ω) : max x ∈ Ω µ ( x ) < ∗ − − p ∗ − , lim s → k sg ( · , s ) − µ k L ∞ (Ω) = 0 . Remark 2.9.
Notice that (G) and ( e g ) imply that lim s → k sg ( · , s ) k L ∞ (Ω) = 0 , i.e. µ ≡ incondition ( e g ). Indeed, we first observe that the fact that g ≥ implies that µ ( x ) ≥ µ ( x ) − sg ( x, s ) ≥ −k sg ( · , s ) − µ k L ∞ (Ω) ∀ ( x, s ) ∈ Ω × (0 , s ) . Passing to the limit as s → leads to µ ≥ . On the other hand, we know that for every ε > there exists s ε ∈ (0 , such that sg ( x, s ) ≥ µ ( x ) − ε ∀ ( x, s ) ∈ Ω × (0 , s ε ) . Let us assume by contradiction that there exists x ∈ Ω such that µ ( x ) > . Then we choose ε = µ ( x )2 and we obtain G ( s ) ≥ g ( x , s ) ≥ εs ∀ s ∈ (0 , s ε ) . This contradicts hypothesis (G).The result reads as follows:
Theorem 2.10.
Let g : Ω × (0 , + ∞ ) → R be a continuous function. For s ∈ (0 , and p ∈ (1 , ∗ − , assume that g satisfies (G) and ( e g ) . Let us also consider the function f : [0 , + ∞ ) → [0 + ∞ ) defined by f ( s ) = s p for all s ≥ . Then, there exists λ > such that there exists atleast a finite energy solution u λ to ( P λ ) for every λ > λ satisfying lim λ → + ∞ k u λ k L ∞ (Ω) = 0 . Remark 2.11.
We point out that Theorem 2.10 is valid for a very wide class of functions g .For instance, if g ( x, s ) = µ ( x ) h ( s ) ∀ ( x, s ) ∈ Ω × [0 , + ∞ ) , where (cid:12) µ ∈ C (Ω) and h : [0 , + ∞ ) → [0 , + ∞ ) is continuous (also at s = 0 ), then g satisfies (G)and ( e g ). On the other hand, a prototypical example of function g singular at s = 0 satisfyingthe conditions (G) and ( e g ) is g ( x, s ) = µ ( x ) s γ ∀ ( x, s ) ∈ Ω × (0 , + ∞ ) where γ ∈ (0 , , (cid:12) µ ∈ C (Ω) .The last theorem of the section will be concerned with the following problem:( H ) − ∆ u + g ( x, u ) |∇ u | = h ( x ) , x ∈ Ω ,u > , x ∈ Ω ,u = 0 , x ∈ ∂ Ω . The result provides a necessary condition for the existence of solution to ( H ). In order to proveit, we will assume, for fixed τ > , that g ( x, u ) is sufficiently large near ∂ Ω in the sense of thefollowing condition:( g ) ∃ ω ⊂⊂ Ω , s ∈ (0 ,
1) : sg ( x, s ) ≥ τ a.e. x ∈ Ω \ ω, ∀ s ∈ (0 , s ) . The theorem states the following:
Theorem 2.12.
Let h ∈ L (Ω) and let g : Ω × (0 , + ∞ ) → R be a Carathéodory functionsatisfying (2.1) and ( g ) for some τ > . Then, every solution u to ( H ) satisfies Z Ω | h ( x ) | u = + ∞ . A priori estimates
We begin the section with two Liouville type results which will be the key points for provinga priori estimates. The first of them is the following.
Lemma 3.1.
Let p > and let h : (0 , + ∞ ) → [0 , + ∞ ) be a continuous function satisfying sh ( s ) ≤ σ ∀ s > for some σ ∈ (0 , . Let us assume that there exists α > such that the function ψ : [0 , + ∞ ) → [0 , + ∞ ) , defined by ψ ( s ) = Z s e − R tα h ( r ) dr dt ∀ s ≥ , satisfies that s ψ ′ ( s ) s p ψ ( s ) ∗ − is decreasing for all s > . Then, the following problem (3.1) − ∆ u + h ( u ) |∇ u | = u p , x ∈ X,u > , x ∈ X,u = 0 , x ∈ ∂X, admits no solutions in H loc ( X ) ∩ C ( X ) , where X denotes either R N or R N + . Remark 3.2.
We stress that Lemma 3.1 includes the particular case h ( s ) = σs with σ < ∗ − − p ∗ − .Indeed, in this case, it is easy to check that ψ ′ ( s ) s p ψ ( s ) ∗− = cs p − σ − (1 − σ )(2 ∗ − for some constant c > .Thus, it is decreasing if, and only if, σ < ∗ − − p ∗ − . Moreover, ∗ − − p ∗ − < , so sh ( s ) = σ < forall s > . Proof of Lemma 3.1.
Arguing by contradiction, assume that there exists a solution u ∈ H loc ( X ) ∩ C ( X ) to (3.1). Straightforward computations imply that v = ψ ( u ) ∈ H loc ( X ) ∩ C ( X ) satisfies(3.2) − ∆ v = ϕ ( v ) , x ∈ X,v > , x ∈ X,v = 0 , x ∈ ∂X, where ϕ ( t ) = ψ ′ ( ψ − ( t )) ψ − ( t ) p for all t ∈ Im ( ψ ) . Moreover, classical elliptic regularity theoryimplies that v ∈ C ( X ) . We claim now that, actually, problem (3.2) admits no solutions v ∈ C ( X ) ∩ C ( X ) . LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 11
Indeed, one can easily deduce that ψ ( s ) ≥ α σ (cid:0) s − σ − α − σ (cid:1) − σ ∀ s > α. Hence, lim s → + ∞ ψ ( s ) = + ∞ . In consequence, the function ϕ is defined in [0 , + ∞ ) . On theother hand, it is clear that the function t ϕ ( t ) t ∗− is decreasing for all t > . Therefore, in case X = R N , the claim follows from [9, Theorem 3].Notice also that lim t → ϕ ( t ) t = 0 and ϕ ( t ) ≥ cψ − ( t ) p − σ for every t > and for some c > .Hence, the claim also holds true in case X = R N + by virtue of [20, Theorem 1.3].In any case, we have arrived to a contradiction. The proof is now completed. (cid:3) We present here the second Liouville type result which is valid for supersolutions.
Lemma 3.3.
Let p > and σ ≤ N − ( N − p . Then, the following problem (3.3) − ∆ u + σ |∇ u | u = u p , x ∈ X,u > , x ∈ X,u = 0 , x ∈ ∂X, admits no supersolutions in H loc ( X ) ∩ C ( X ) provided X = R N . On the other hand, if weassume that σ ≤ N +1 − ( N − p , then problem (3.3) with X = R N + admits no supersolutions in H loc ( X ) ∩ C ( X ) . Remark 3.4.
Note that N +1 − ( N − p < N − ( N − p < ∗ − − p ∗ − < , so the smallness conditionson σ in Remark 3.2 and in Lemma 3.3 are gradually more restrictive. We also stress that suchconditions on σ in Lemma 3.3 are sharp. Indeed, if σ > N − ( N − p (resp. σ > N +1 − ( N − p ),then one can find explicit supersolutions to (3.3) for X = R N , see [33] (resp. X = R N + , see [10]). Proof of Lemma 3.3.
Reasoning by contradiction, assume that there exists u ∈ H loc ( X ) ∩ C ( X ) a supersolution to (3.3). Then, there is a constant c > such that v = cu − σ ∈ H loc ( X ) ∩ C ( X ) is a supersolution to − ∆ v = v p − σ − σ , x ∈ X,v > , x ∈ X,v = 0 , x ∈ ∂X. Hence, if σ ≤ N − ( N − p , then p − σ − σ ≤ NN − , so in case X = R N we arrived to a contractionwith [33, Theorem 2.1]. On the other hand, if σ < N +1 − ( N − p , then p − σ − σ < N +1 N − , so we haveagain a contradiction with [10, Theorem 3.1]. (cid:3) Let t ≥ , λ > , f : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function satisfying ( f ∗ ) for some p > and g : Ω × (0 , + ∞ ) → [0 , + ∞ ) be a Carathéodory function satisfying ( g ) for some σ ∈ (0 , . From now and up to the end of the section, we will restrict ourselves to g ≥ and wewill also impose diverse upper bounds on p . However, we will show in Section 4 that, in mostcases, those restrictions can be relaxed for proving existence of solution. Let us consider the following auxiliary problem:( P t ) − ∆ u + g ( x, u ) |∇ u | = λf ( u ) + tu σ , x ∈ Ω ,u > , x ∈ Ω ,u = 0 , x ∈ ∂ Ω . Note that, in the term tu σ , the exponent σ is the same number that appears in condition ( g ).We will derive a priori estimates on the solutions to ( P t ) which will provide the existence ofsolution to ( P λ ).Next proposition gives an a priori estimate on the parameter t in problem ( P t ). Proposition 3.5.
Let λ > , f : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function and g : Ω × (0 , + ∞ ) → [0 , + ∞ ) be a Carathéodory function. For p > and σ ∈ (0 , , assume that f satisfies ( f ∗ ) and g satisfies ( g ) . Then, there exists t > such that problem ( P t ) admits nosolution for any t > t .Proof. Let u be a solution to ( P t ) for some t > . For a fixed smooth open set ω ⊂⊂ Ω , let λ be the principal eigenvalue to the homogeneous Dirichlet eigenvalue problem in ω , and let ϕ beany positive associated eigenfunction, i.e. λ and ϕ satisfy − ∆ ϕ = λ ϕ , x ∈ ω,ϕ > , x ∈ ωϕ = 0 , x ∈ ∂ω. If we extend ϕ ≡ in Ω \ ω , then the function φ = ϕ u σ belongs to H (Ω) ∩ L ∞ (Ω) and hascompact support. Taking φ as test function in ( P t ) (which is allowed by virtue of Remark 2.2)we obtain(3.4) Z Ω ∇ u ∇ ϕ u σ − σ Z Ω ϕ |∇ u | u σ +1 + Z Ω g ( x, u ) ϕ |∇ u | u σ = λ Z Ω f ( u ) u σ ϕ + t Z Ω ϕ . On the one hand, it is clear by ( g ) that − σ Z Ω ϕ |∇ u | u σ +1 + Z Ω g ( x, u ) ϕ |∇ u | u σ ≤ . Thus, using also ( f ∗ ) we deduce from (3.4) that(3.5) t Z Ω ϕ + λ Z Ω u p − σ ϕ ≤ Z Ω ∇ u ∇ ϕ u σ . On the other hand, let us denote as ν the exterior normal unit vector to ∂ω . Then, Hopf’slemma implies that ν ∇ ϕ < on ∂ω . Hence, integration by parts in R ω ( − ∆ ϕ ) u − σ and Young’sinequality yield Z Ω ∇ u ∇ ϕ u σ = λ Z ω ϕ u − σ − σ + Z ∂ω u − σ − σ ν ∇ ϕ < λ Z ω ϕ u − σ − σ ≤ λ Z Ω u p − σ ϕ + C. In sum, from (3.5) we deduce that t ≤ t for some t > , as we wanted to prove. (cid:3) Next lemma provides a summability property that the solutions to ( P t ) satisfy. The interestingpoint is that such a property becomes better as the singularity of g becomes stronger. LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 13
Lemma 3.6.
Let λ > , σ ∈ (0 , , f : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function and g : Ω × (0 , + ∞ ) → [0 , + ∞ ) be a Carathéodory function satisfying (2.1) . For some a > , p ≥ and τ ∈ [0 , , assume that f ( s ) ≤ as p ∀ s ≥ ,sg ( x, s ) ≥ τ a.e. x ∈ Ω , ∀ s > . Let u ∈ H loc (Ω) ∩ L ∞ (Ω) be a solution to ( P t ) for some t ≥ . Then, u γ ∈ H (Ω) for every γ > max { − σ, − τ } .Proof. First of all, recall that Lemma 5.2 in the Appendix below implies that u ∈ C (Ω) . Now,for any α > , let us consider the function v = u α ∈ H loc (Ω) ∩ C (Ω) . It is easy to see that v satisfies − ∆ v = (1 − α − v α g ( x, v α )) |∇ v | αv + αλv α − α f ( v α ) + αtv α − σα ≤ (1 − α − τ ) |∇ v | αv + αλav α − pα + αtv α − σα , x ∈ Ω . Moreover, if we take α ≥ max { − σ, − τ } , then(3.6) − ∆ v ≤ C, x ∈ Ω , for some constant C > .For every ε, β > , let us now consider the function φ = ( v β − ε ) + . It is clear that φ ∈ H loc (Ω) ∩ C (Ω) . Furthermore, the continuity of v up to ∂ Ω implies that φ has compact supportin Ω . In sum, φ ∈ H (Ω) ∩ C (Ω) and has compact support. Therefore, even though v might notbelong to H (Ω) , it follows from Remark 2.2 that one can take φ as test function in (3.6). Thus,we obtain Z Ω χ { v β ≥ ε } |∇ v | v − β ≤ C, for another constant C > independent of ε . Now we let ε tend to zero and, by virtue of Fatou’slemma, we deduce Z Ω |∇ v | v − β ≤ C. Taking into account that |∇ v | v − β = C (cid:12)(cid:12)(cid:12) ∇ v β (cid:12)(cid:12)(cid:12) = C (cid:12)(cid:12)(cid:12) ∇ u α β (cid:12)(cid:12)(cid:12) , then we have proved that u α β ∈ H (Ω) for every α ≥ max { − σ, − τ } and every β > . Thisproves the result. (cid:3) In the following result we prove a priori estimates on the solutions to ( P t ) if f satisfies ( f ∗ )and ( f ∞ ) and g satisfies ( g ∗ ) and ( g ∞ ). In the proof we adapt the blow-up method due to [25]. Proposition 3.7.
Let λ > , f : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function and g : Ω × (0 , + ∞ ) → [0 , + ∞ ) be a Carathédory function. For δ, τ ≥ , σ > and p ∈ (1 , ∗ − , assumethat f satisfies ( f ∗ ) and ( f ∞ ) , and that g satisfies ( g ∗ ) and ( g ∞ ) . Then, there exists C > such that k u k L ∞ (Ω) ≤ C for every solution u to ( P t ) for all t ∈ [0 , t ] , where t > is given byProposition 3.5. Proof.
Reasoning by contradiction, we assume that there exist two sequences { t n } ⊂ [0 , t ] and { u n } such that u n is a solution to ( P t n ) for all n and k u n k L ∞ (Ω) → + ∞ as n → + ∞ . By virtueof Lemma 5.4, we may consider a sequence { x n } ⊂ Ω satisfying k u n k L ∞ (Ω) = u n ( x n ) ∀ n, x n → x ∈ Ω , up to a subsequence . We divide the rest of the proof into two parts. In the first of them we consider the case x ∈ Ω ,while the second one is devoted to the case x ∈ ∂ Ω . In turn, we divide each part into severalsteps. CASE 1) x ∈ Ω .Step 1.1) Scaling. Denoting d = dist ( x , ∂ Ω) / > and η n = k u n k − p − L ∞ (Ω) , we define v n : B d/η n (0) → [0 , + ∞ ) by v n ( y ) = η p − n u n ( x n + η n y ) ∀ y ∈ B d/η n (0) . Therefore, v n ∈ H ( B d/η n (0)) ∩ L ∞ ( B d/η n (0)) and satisfies the equation(3.7) − ∆ v n + u n g n ( y, u n ) |∇ v n | v n = λv pn f ( u n ) u pn + t n η p − σ ) p − n v σn , y ∈ B d/η n (0) , where g n ( y, s ) = g ( x n + η n y, s ) for a.e. y ∈ B d/η n (0) and s > . Moreover, k v n k L ∞ ( B d/ηn (0)) = v n (0) = 1 . Our aim now is to pass to the limit in (3.7). In the next step we will prove the apriori estimates that will provide such a limit.
Step 1.2) A priori estimates.
Let us fix
R > and denote ω = B R (0) . It is clear that ω ⊂ B d/η n (0) for every n sufficientlylarge, so v n satisfies the equation (3.7) in ω and k v n k L ∞ ( ω ) = 1 for n large. Of course, the samething happens in B R (0) . Notice also that, if δ = 0 , then Lemma 3.6 implies that u γn ∈ H (Ω) for all γ ∈ (cid:16) max { − σ, − τ } , − σ i = (cid:0) − τ , − σ (cid:3) , where this last equality follows from τ < σ incondition ( g ∗ ). Therefore, v γn ∈ H ( B R (0)) for all γ ∈ (cid:0) − τ , − σ (cid:3) . This fact, together withconditions ( g ∗ ) and ( f ∗ ), allow to apply Lemma 5.5 in the Appendix below to deduce that thereexist C > , α ∈ (0 , such that k v n k C ,α ( ω ) ≤ C for every n large enough. As a consequence, there exists v ∈ C ( ω ) such that, up to a subsequence, v n → v uniformly in ω. Observe that k v k L ∞ ( ω ) = 1 so, in particular, v .On the other hand, let us consider a function φ ∈ C c ( B R (0)) such that ≤ φ ≤ in B R (0) and φ ≡ in ω . Now we multiply both sides of (3.7) by v n φ ∈ H ( B R (0)) ∩ L ∞ ( B R (0)) andintegrate by parts, obtaining Z B R (0) |∇ v n | φ + 2 Z B R (0) v n φ ∇ v n ∇ φ ≤ C, where we have used that g ≥ , k v n k L ∞ ( B R (0)) = 1 and, one more time, condition ( f ∗ ). Hence,by Young’s inequality we easily deduce that Z ω |∇ v n | ≤ C Z B R (0) |∇ φ | v n + 1 ! ≤ C. LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 15
That is to say, k v n k H ( ω ) ≤ C , and then, up to a subsequence, v n ⇀ v weakly in H ( ω ) . We will prove next that, for all ω ⊂⊂ ω , v n is bounded from below in ω by a positiveconstant independent of n . The approach by comparison due to [32] is valid here. Indeed, it isstraightforward to see that the function w n = v − σn − σ ∈ H ( ω ) ∩ L ∞ ( ω ) satisfies Z ω ∇ w n ∇ φ = λ Z ω v p − σn f ( u n ) u pn φ + Z ω ( σ − u n g n ( y, u n )) |∇ v n | φv σ +1 n + t n η p − σ ) p − n Z ω φ for all φ ∈ C c ( ω ) . Therefore, using conditions ( g ∗ ) and ( f ∗ ), we derive Z ω ∇ w n ∇ φ ≥ λ Z ω v p − σn φ for all ≤ φ ∈ C c ( ω ) . On the other hand, let z n ∈ H ( ω ) ∩ C ( ω ) be the unique solution to ( − ∆ z n = λv p − σn , y ∈ ω,z n = 0 , y ∈ ∂ω. It is clear that z n → z uniformly in ω and z n ⇀ z weakly in H ( ω ) for some z ∈ H ( ω ) ∩ C ( ω ) .In consequence, z satisfies ( − ∆ z = λv p − σ , y ∈ ω,z = 0 , y ∈ ∂ω. Since v (cid:13) in ω , the strong maximum principle implies that, for every ω ⊂⊂ ω , there exists c > such that z ≥ c in ω . Besides, by comparison, w n ≥ z n in ω . Hence, the uniform convergenceof { z n } implies that, for every ε > , we may take n large enough so that v − σn ≥ (1 − σ )( z − ε ) in ω . In sum, by choosing ε = c we conclude that(3.8) ∀ ω ⊂⊂ ω, ∃ c ω > v n ≥ c ω , y ∈ ω , ∀ n large.From the previous estimates, it is straightforward to prove that { ∆ v n } is bounded in L loc ( ω ) .Then, [12] implies that, passing to a subsequence, ∇ v n → ∇ v, y ∈ ω. We are ready now to pass to the limit in (3.7).
Step 1.3) Passing to the limit.
We already know that v n ≥ c ω > in ω ⊂⊂ ω . Moreover, k u n k L ∞ ( ω ) → + ∞ . Therefore, u n → + ∞ locally uniformly in ω . Then, by ( g ∞ ) we deduce that | u n g n ( y, u n ) − µ ( x n + η n y ) | → locally uniformly in ω . In consequence, the continuity of µ yields u n g n ( y, u n ) → µ ( x ) locally uniformly in ω. In sum, we have that ≤ u n g n ( y, u n ) |∇ v n | v n → µ ( x ) |∇ v | v pointwise in ω. Furthermore, it follows from conditions ( f ∞ ) and ( f ∗ ) and from the Dominated ConvergenceTheorem that Z ω v pn f ( u n ) u pn φ → Z ω v p φ ∀ φ ∈ C c ( ω ) . Let us take φ ∈ C c ( ω ) such that φ ≥ as test function in the weak formulation of (3.7). Byvirtue of Fatou’s lemma and using the convergences that we have proved, it is immediate to showthat Z ω ∇ v ∇ φ + µ ( x ) Z ω |∇ v | v φ ≤ λ Z ω v p φ. If we take now vφv n ∈ H ( ω ) ∩ L ∞ ( ω ) as test function in (3.7), we obtain Z ω ∇ v n ∇ vv n φ − Z ω v |∇ v n | v n φ + Z ω vv n ∇ v n ∇ φ + Z ω u n g n ( y, u n ) v |∇ v n | v n φ = λ Z ω v p − n v f ( u n ) u pn φ + t n η p − σ ) p − n Z ω vφv − σn . Observe that ≤ (1 − σ ) v |∇ v n | φv n ≤ (1 − u n g n ( y, u n )) v |∇ v n | φv n → (1 − µ ( x )) |∇ v | v φ. Then, again by Fatou’s lemma we derive Z ω ∇ v ∇ φ + µ ( x ) Z ω |∇ v | v φ ≥ λ Z ω v p φ. That is to say, v ∈ H ( ω ) ∩ C ( ω ) satisfies − ∆ v + µ ( x ) |∇ v | v = λv p , y ∈ ω. Step 1.4) Conclusion.
Since ω = B R (0) for arbitrary R > , then a standard diagonal argument (see [25]) impliesthat v is well-defined in R N , it belongs to H loc ( R N ) ∩ C ( R N ) and it satisfies − ∆ v + µ ( x ) |∇ v | v = λv p , y ∈ R N . Furthermore, it is straightforward to check that the function w = λ p − v satisfies − ∆ w + µ ( x ) |∇ w | w = w p , y ∈ R N . This is impossible by virtue of Lemma 3.1 (see also Remark 3.2).
CASE 2) x ∈ ∂ Ω .Step 2.1) Scaling. Recall that we are assuming that there exist sequences { t n } ⊂ [0 , t ] , { u n } and { x n } ⊂ Ω suchthat u n is a solution to ( P t n ) for all n , k u n k L ∞ (Ω) = u n ( x n ) → + ∞ as n → + ∞ and x n → x for some x ∈ ∂ Ω . Taking advantage of the smoothness of ∂ Ω , we are allowed to perform aconvenient change of coordinates in such a way that u n is a solution to a similar problem exceptthat ∂ Ω becomes flat near x (see Lemma 5.1 in the Appendix below for the detailed proof). Inother words, we may assume without loss of generality that u n is a solution to ( R t n ) for all n ,where( R t ) − div ( M ( x ) ∇ v ) + b ( x ) ∇ v + g ( x, v ) M ( x ) ∇ v ∇ v = λf ( v ) + tv σ , x ∈ Ω ,v > , x ∈ Ω ,v = 0 , x ∈ Γ , LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 17 with Ω ⊂ R N + , ∅ 6 = Γ ⊂ ∂ Ω ∩ ∂ R N + being open as a subset of ∂ R N + and connected, M ∈ C (Ω) N × N being uniformly elliptic, and b ∈ C (Ω) N .Since Γ is open in ∂ R N + , then d n = dist ( x n , ∂ Ω) = dist ( x n , Γ) = x n,N for all n large enough.Arguing as in the previous case, we define v n ( y ) = η p − n u n ( x n + η n y ) ∀ y ∈ Ω n ∪ Γ n , where η n = k u n k − p − L ∞ (Ω) , ∈ Ω n = B d/η n (0) ∩{ y N > − d n /η n } and Γ n = B d/η n (0) ∩{ y N = − d n /η n } for some d > . It is not difficult to see that v n is well-defined for all n large enough and itsatisfies − div ( M n ( y ) ∇ v n ) + b n ( y ) ∇ v n + g n ( y, u n ) u n M n ( y ) ∇ v n ∇ v n v n = λv pn f ( u n ) u pn + η p − σ ) p − n t n v σn , y ∈ Ω n ,v n > , x ∈ Ω n ,v n = 0 , y ∈ Γ n , where M n ( y ) = M ( x n + η n y ) , b n ( y ) = η n b ( x n + η n y ) and g n ( y, · ) = g ( x n + η n y, · ) .Now, if { d n /η n } is unbounded, we can extract a subsequence such that d n /η n → + ∞ as n → + ∞ . In this case, S n ∈ N Ω n = R N , so we can argue as in the case x ∈ Ω without relevantchanges and arrive to a contradiction.Let us assume now that { d n /η n } is bounded. Then, up to a (not relabeled) subsequence, d n /η n → κ for some κ ≥ . Thus, S n ∈ N Ω n = { y N > κ } . Let us prove the estimates in this newsituation. Step 2.2) A priori estimates.
Let us denote T n = (0 , ..., , d n /η n ) ∈ R N , Ω ′ n = B d/η n ( T n ) ∩ R N + , Γ ′ n = B d/η n ( T n ) ∩ ∂ R N + . Observe that S n ∈ N Ω ′ n = R N + and S n ∈ N Γ ′ n = ∂ R N + . Let us define w n : Ω ′ n ∪ Γ ′ n → [0 , + ∞ ) by w n ( y ) = v n ( y − T n ) ∀ y ∈ Ω ′ n ∪ Γ ′ n . Obviously, w n satisfies(3.9) − div ( M ′ n ( y ) ∇ w n ) + b ′ n ( y ) ∇ w n + g ′ n y, w n η p − n ! η p − n M ′ n ( y ) ∇ v n ∇ v n = λη pp − n f w n η p − n ! + η p − σ ) p − n t n w σn , y ∈ Ω ′ n ,w n > , x ∈ Ω ′ n ,w n = 0 , y ∈ Γ ′ n , where M ′ n ( y ) = M n ( y − T n ) , b ′ n ( y ) = b n ( y − T n ) and g ′ n ( y, · ) = g n ( y − T n , · ) . We also have that k w n k L ∞ (Ω ′ n ) = w n ( T n ) = 1 for all n .Let us denote T ∞ = (0 , ..., , κ ) ∈ R N and, for R > κ , let us consider the sets ω = B R ( T ∞ ) ∩ R N + , β = B R ( T ∞ ) ∩ ∂ R N + ω = B R ( T ∞ ) ∩ R N + , β = B R ( T ∞ ) ∩ ∂ R N + . It is easy to see that, for all n large enough, T n ∈ ω ⊂ ω ⊂ Ω ′ n , ∈ β ⊂ β ⊂ Γ ′ n , ω ⊆ ω ∪ β . In particular, w n satisfies problem (3.9) by changing Ω ′ n with ω and Γ ′ n with β , and further, k w n k L ∞ ( ω ) = 1 for all n large enough. Thus, it follows again from Lemma 5.5 in the Appendix(thanks to conditions ( g ∗ ) and ( f ∗ ) and to Lemma 3.6) that there exist C > , α ∈ (0 , suchthat k w n k C ,α ( ω ) ≤ C for every n large enough. As a consequence, there exists v ∈ C ( ω ) such that, up to a subsequence, w n → v uniformly in ω . In particular, v = 0 on β .On the other hand, from the Hölder estimate one can also deduce that κ > . Indeed, | w n ( T n ) − w n (0) | ≤ C | T n | α = C ( d n /η n ) α → Cκ α . Next, an estimate on { w n } in H loc ( ω ) can be proven as in CASE 1, so that w n ⇀ v weaklyin H loc ( ω ) , up to a subsequence. Moreover, the same arguments are valid to prove that { w n } satisfies also (3.8) and, furthermore, that ∇ w n → ∇ v a.e. in ω . Step 2.3) Passing to the limit.
We can now pass to the limit as in CASE 1 and deduce that v ∈ H loc ( ω ) ∩ C ( ω ) is a solutionto the following problem: − div ( M ( x ) ∇ v ) + µ ( x ) M ( x ) ∇ v ∇ vv = λv p , y ∈ ω,v > , y ∈ ω,v = 0 , y ∈ β. Actually, as R is arbitrary, the diagonal argument (see [25]) implies that v ∈ H loc ( R N + ) ∩ C (cid:16) R N + (cid:17) and it satisfies − div ( M ( x ) ∇ v ) + µ ( x ) M ( x ) ∇ v ∇ vv = λv p , y ∈ R N + ,v > , y ∈ R N + ,v = 0 , y ∈ ∂ R N + . Step 2.4) Conclusion.
Observe that, if we denote M ( x ) = ( m ij ) for i, j = 1 , ..., N , then the previous equation maybe written as N X i,j =1 m ij ∂ v∂y i ∂y j + µ ( x ) v N X i,j =1 m ij ∂v∂y i ∂v∂y j = λv p , y ∈ R N + . Since i, j commute in both ∂ v∂y i ∂y j and ∂v∂y i ∂v∂y j , then a simple change of coordinates (see theconclusion of Case 1 in Section 2 of [25] for the details) leads to finding a solution w ∈ H loc ( R N + ) ∩ C (cid:16) R N + (cid:17) to − ∆ w + µ ( y ) |∇ w | w = w p , y ∈ R N + ,w > , y ∈ R N + ,w = 0 , y ∈ ∂ R N + , for some y ∈ ∂ Ω . This contradicts Lemma 3.1 (see also Remark 3.2). The proof is concluded. (cid:3) LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 19
The following result, as Proposition 3.7, provides a priori estimates on the solutions to ( P t ).The difference lies on the fact that we do not impose the limit conditions at infinity ( g ∞ ) and( f ∞ ) at the expense of making a stronger restriction on σ, p . Proposition 3.8.
Let λ > , f : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function and g : Ω × (0 , + ∞ ) → [0 , + ∞ ) be a Carathéodory function. For δ, τ ≥ , σ > and p ∈ (cid:16) , N +1 N − (cid:17) , assumethat f satisfies ( f ∗ ) and g satisfies ( g ∗ ) . Assume in addition that σ ≤ N +1 − ( N − p . Then, thereexists C > such that k u k L ∞ (Ω) ≤ C for every solution u to ( P t ) for all t ∈ [0 , t ] , where t > is given by Proposition 3.5.Proof. The proof is very similar to that of Proposition 3.7. Here we give only a sketch.Arguing by contradiction, we assume that there exist two sequences { t n } ⊂ [0 , t ] and { u n } such that u n is a solution to ( P t n ) for all n and k u n k L ∞ (Ω) → + ∞ as n → + ∞ . We also considera sequence { x n } ⊂ Ω satisfying k u n k L ∞ (Ω) = u n ( x n ) ∀ n, x n → x ∈ Ω , up to a subsequence ; this can be done by virtue of Lemma 5.4. We denote η n = k u n k − p − L ∞ (Ω) . Let us assume that x ∈ ∂ Ω (we omit the simpler case x ∈ Ω ). Arguing as in the proof of Proposition 3.7, weassume without loss of generality that u n is a solution to ( R t n ) for all n , with Ω = V ⊂ R N + and Γ ⊂ ∂ Ω ∩ ∂ R N + , being Γ open as a subset of ∂ R N + and connected. Thus, there exists a sequenceof bounded domains { Ω n } satisfying, for every n , that ∈ Ω n , x n + η n y ∈ Ω for all y ∈ Ω n and S n ∈ N Ω n = X , where X may be either R N or { y N > κ } for some κ ≥ .In any case, we define v n : Ω n → R by v n ( y ) = η p − n u n ( x n + η n y ) ∀ y ∈ Ω n . It is easy to check that v n satisfies the equation − div ( M n ( y ) ∇ v n ) + b n ( y ) ∇ v n + u n g n ( y, u n ) M n ( y ) ∇ v n ∇ v n v n = λv pn f ( u n ) u pn + η p − σ ) p − n t n v σn , y ∈ Ω n , where M n ( y ) = M ( x n + η n y ) , b n ( y ) = η n b ( x n + η n y ) and g n ( y, · ) = g ( x n + η n y, · ) . From theprevious equation one obtains the same local estimates as in Proposition 3.7. However, in thiscase one cannot pass to the limit directly in the equation. We overcome this issue by deducingfrom sg n ( x, s ) ≤ σ , M n ( y ) ∇ v n ∇ v n ≥ and f ( u n ) ≥ u pn the inequality − div ( M n ( y ) ∇ v n ) + b n ( y ) ∇ v n + σ M n ( y ) ∇ v n ∇ v n v n ≥ λv pn , y ∈ Ω n . In fact, one can pass to the limit in the previous inequality as in Proposition 3.7 (but usingFatou’s lemma only once). Applying after that a convenient change of coordinates, we finda supersolution v ∈ H loc ( X ) ∩ C ( X ) to (3.3), where either X = R N or X = R N + . This is acontradiction with Lemma 3.3. (cid:3) Next proposition provides similar estimates as Proposition 3.7 and Proposition 3.8. Thenovelty is that g satisfies now a limit condition at infinity only in a neighborhood of ∂ Ω . Thismeans that we need to impose stronger restrictions on σ, p than in Proposition 3.7, but milderthan in Proposition 3.8. Proposition 3.9.
Let λ > , f : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function and g : Ω × (0 , + ∞ ) → [0 , + ∞ ) be a Carathéodory function. For δ, τ ≥ , σ > and p ∈ (cid:16) , NN − (cid:17) , assumethat f satisfies ( f ∗ ) and ( f ∞ ) , and that g satisfies ( g ∗ ) and ( f g ∞ ) . Assume in addition that σ ≤ N − ( N − p . Then, there exists C > such that k u k L ∞ (Ω) ≤ C for every solution u to ( P t ) for all t ∈ [0 , t ] , where t > is given by Proposition 3.5.Proof. We argue similarly as for Proposition 3.7 and Proposition 3.8, so we provide only themain ideas.Assume by contradiction that there exist two sequences { t n } ⊂ [0 , t ] and { u n } such that u n is a solution to ( P t n ) for all n and k u n k L ∞ (Ω) → + ∞ as n → + ∞ . Thanks to Lemma 5.4, wemay also consider a sequence { x n } ⊂ Ω satisfying k u n k L ∞ (Ω) = u n ( x n ) ∀ n, x n → x ∈ Ω up to a subsequence . Suppose that x ∈ Ω . Since x might belong to ω (where we know nothing about the asymptoticbehavior of g at infinity), we cannot proceed as in the proof of Proposition 3.7. Nevertheless,scaling u n conveniently and using ( g ∗ ) and ( f ∗ ) we may argue as in the proof of Proposition 3.8to find a supersolution < v ∈ H loc ( R N ) ∩ C ( R N ) to − ∆ v + σ |∇ v | v = v p , y ∈ R N . This is a contradiction with Lemma 3.3.On the other hand, if x ∈ ∂ Ω , then we may take advantage of ( f g ∞ ) to obtain, using also ( f ∞ )and arguing as in Proposition 3.7, a solution v ∈ H loc ( X ) ∩ C ( X ) to (3.3) for σ = µ ( y ) and some y ∈ Ω , and either X = R N or X = R N + . This contradicts Lemma 3.1 (see also Remark 3.2). (cid:3) Next result provides an estimate for the solutions to problem ( P λ ) whose dependence on λ isexplicit. As a consequence, it is shown that the norm of the solutions to problem ( P λ ), if theyexist, becomes arbitrarily small as λ tends to infinity. Proposition 3.10.
Let f : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function and g : Ω × (0 , + ∞ ) → [0 , + ∞ ) be a Carathéodory function. For δ = 0 , τ ≥ , σ > and p ∈ (cid:16) , N +1 N − (cid:17) , assume that f satisfies ( f ∗ ) and g satisfies ( g ∗ ) . Assume in addition that σ ≤ N +1 − ( N − p . Then, there exists C > such that λ p − k u k L ∞ (Ω) ≤ C for every solution u to ( P λ ) for all λ > .Proof. Arguing again as in the proof of Proposition 3.7, assume that there exist two sequences { λ n } ⊂ [0 , + ∞ ) and { u n } such that u n is a solution to ( P λ n ) for all n and k z n k L ∞ (Ω) → + ∞ as n → + ∞ , where z n = λ p − n u n . It is easy to see that z n satisfies − ∆ z n + λ − p − n g (cid:18) x, λ − p − n z n (cid:19) |∇ z n | = λ pp − n f (cid:18) λ − p − n z n (cid:19) , x ∈ Ω . Since z n ∈ C (Ω) by virtue of Lemma 5.4, then we may take { x n } ⊂ Ω such that z n ( x n ) = k z n k L ∞ (Ω) for all n . Let x ∈ Ω be such that, passing to a subsequence if necessary, x n → x ∈ ∂ Ω (the case x ∈ Ω is analogous, so we omit it). Arguing as in the proof of Proposition 3.7 LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 21 (Case 2), we may assume without loss of generality that v n ∈ H (Ω n ) ∩ L ∞ (Ω n ) , defined by v n ( y ) = η p − n z n ( x n + η n y ) for all y ∈ Ω n , satisfies(3.10) − div ( M n ( y ) ∇ v n )+ b n ( y ) ∇ v n + v n L n g n (cid:18) y, v n L n (cid:19) M n ( y ) ∇ v n ∇ v n v n = L pn f (cid:18) v n L n (cid:19) , y ∈ Ω n , where η n = k z n k − p − L ∞ (Ω) , L p − n = λ n η n and Ω n , Γ n , M n , b n , g n are as in the proof of Proposition 3.7(Case 2). From (3.10), and using conditions ( f ∗ ) and ( g ∗ ), one can prove the same local estimateson { v n } as in the proof of Proposition 3.7. Moreover, again by ( f ∗ ) and ( g ∗ ) we deduce that − div ( M n ( y ) ∇ v n ) + b n ( y ) ∇ v n + σ M n ( y ) ∇ v n ∇ v n v n ≥ v pn , y ∈ Ω n . Now we pass to the limit in the previous inequality by using Fatou’s lemma once and obtaina supersolution v ∈ H loc ( X ) ∩ C ( X ) to (3.3), where either X = R N or X = R N + . This is acontradiction by virtue of Lemma 3.3. (cid:3) In the last result of this section we impose stronger restrictions on f and g to obtain an apriori estimate for the solutions to ( P λ ) similar to that in Proposition 3.10. The advantage isthat p can be chosen near ∗ − . Proposition 3.11.
Let g : Ω × (0 , + ∞ ) → [0 , + ∞ ) be a continuous function. For δ = 0 , τ ≥ , σ > and p ∈ (1 , ∗ − , assume that g satisfies ( g ∗ ) , ( g ∞ ) and ( e g ) . Assume in addition thatthere exists α > such that, for each x ∈ Ω and L > , the function ψ : [0 , + ∞ ) → [0 , + ∞ ) ,defined by ψ ( s ) = Z s e − L R tαL g ( x , rL ) dr dt ∀ s ≥ , satisfies that (3.11) s ψ ′ ( s ) s p ψ ( s ) ∗ − is decreasing for all s > . Lastly, let us consider the function f : [0 , + ∞ ) → [0 , + ∞ ) defined by f ( s ) = s p for all s ≥ .Then, there exists C > such that λ p − k u k L ∞ (Ω) ≤ C for every solution u to ( P λ ) for all λ > .Proof. We may reproduce the proof of Proposition 3.10 up to (3.10), where { v n } satisfies thesame estimates as in the proof of Proposition 3.7. Now, unlike in the proof of Proposition 3.10,we aim to pass to the limit directly in (3.10). In order to do so, let us take a not relabeledsubsequence so that L n → L for some L ∈ [0 , + ∞ ] . Let us assume first that L = 0 . In this case,we can pass to the limit in (3.10) as in the proof of Proposition 3.7 and we obtain a solution v ∈ H loc ( X ) ∩ C ( X ) to (3.3) with σ = µ ( y ) for some y ∈ Ω , where either X = R N or X = R N + .This is a contradiction by virtue of Lemma 3.1 (see also Remark 3.2).Therefore, necessarily L > . Assume now that L = + ∞ . Using ( e g ), we may pass to thelimit again in (3.10) and we find a solution v ∈ H loc ( X ) ∩ C ( X ) to (3.3) with σ = µ ( y ) forsome y ∈ Ω , where either X = R N or X = R N + . One more time, this contradicts Lemma 3.1(see also Remark 3.2).The only remaining possibility is < L < + ∞ . Thanks to the continuity of g , we maypass to the limit in (3.10) once more to obtain a solution v ∈ H loc ( X ) ∩ C ( X ) to (3.1), where h ( s ) = L g (cid:0) y , sL (cid:1) for some y ∈ Ω and either X = R N or X = R N + . Again, this is a contradictionwith Lemma 3.1. (cid:3) Proofs of the main results
We start by proving Theorem 2.3.
Proof of Theorem 2.3.
Let us first define the function ˜ g : Ω × (0 , + ∞ ) → [0 , + ∞ ) by ˜ g ( x, s ) = σ − s − σ g (cid:16) x, s − σ (cid:17) (1 − σ ) s , a.e. x ∈ Ω , ∀ s > . Thanks to ( g ) and ( g ր ), it is clear that ˜ g is nonincreasing in the s variable. Moreover, | ˜ g ( s, x ) | ≤ σ + s − σ g (cid:16) s − σ (cid:17) (1 − σ ) s , a.e. x ∈ Ω , ∀ s > , so it is a bounded function in the x variable.For some ≤ φ ∈ H (Ω) ∩ L ∞ (Ω) with compact support, let us take (1 − σ ) φu σ as test functionin (2.2). Then, if we denote ˜ u = u − σ , we deduce that Z Ω ∇ ˜ u ∇ φ ≤ Z Ω ˜ g ( x, ˜ u ) |∇ ˜ u | φ + Z Ω (1 − σ ) h ( x )˜ u σ − σ φ. Arguing similarly, ˜ v = v − σ satisfies Z Ω ∇ ˜ v ∇ φ ≥ Z Ω ˜ g ( x, ˜ v ) |∇ ˜ v | φ + Z Ω (1 − σ ) h ( x )˜ v σ − σ φ. Moreover, ˜ u, ˜ v ∈ C (Ω) ∩ W ,N loc (Ω) and they satisfy lim sup x → x (˜ u ( x ) − ˜ v ( x )) ≤ for all x ∈ ∂ Ω .At this point one can reproduce the proof of [31, Theorem 2.1] without relevant changes andconclude that ˜ u ≤ ˜ v in Ω . Equivalently, u ≤ v in Ω . (cid:3) Next we prove Theorem 2.12.
Proof of Theorem 2.12.
Let v be a solution to ( H ). Recall that the definition of solution impliesthat there exists γ > such that v γ ∈ H (Ω) . It is easy to see that u = v γ satisfies the equation(4.1) − ∆ u + g γ ( x, u ) |∇ u | = γv γ − h ( x ) , x ∈ Ω , where g γ ( x, s ) = s γ g (cid:16) x, s γ (cid:17) + γ − γs , a.e. x ∈ Ω , ∀ s > . It follows from ( g ) that(4.2) sg γ ( x, s ) ≥ τ + γ − γ a.e. x ∈ Ω \ ω, ∀ s ∈ (0 , s γ ) , where clearly τ + γ − γ > . From now we will denote ρ = τ + γ − γ and s = s γ . LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 23
Recall that there exists c > such that v ≥ c γ in ω . For every ε ∈ (0 , min { s , c } ) , let usdefine the function ϕ ε : [0 , + ∞ ) → [0 , + ∞ ) by ϕ ε ( s ) = s + s ρ − − ε ρ − ( ρ − s ρ , s ≥ ε,sε , ≤ s < ε. Clearly, ϕ ε ( u ) ∈ H (Ω) ∩ L ∞ (Ω) . Hence, taking ϕ ε ( u ) as test function in the weak formulationof (4.1) (this can be done thanks to Remark 2.2 because u ∈ H (Ω) ) we obtain(4.3) Z Ω ϕ ′ ε ( u ) |∇ u | + Z Ω g γ ( x, u ) |∇ u | ϕ ε ( u ) = γ Z Ω v γ − h ( x ) ϕ ε ( u ) . Observe that, from (4.2), it follows that Z Ω g γ ( x, u ) |∇ u | ϕ ε ( u ) = Z ω ∪{ u ≥ s } g γ ( x, u ) |∇ u | ϕ ε ( u ) + Z (Ω \ ω ) ∩{ uε } ρ |∇ u | u − Z { u>ε } ρ |∇ u | u (cid:18) u + u ρ − − ε ρ − ( ρ − u ρ (cid:19) + Z { u ≤ ε } |∇ u | ε . We will now absorb the negative term in (4.5) with the last term in (4.4). Indeed, Z (Ω \ ω ) ∩{ εε } ρ |∇ u | u (cid:18) u + u ρ − − ε ρ − ( ρ − u ρ (cid:19) = − Z [ { εε } ρ |∇ u | u ≤ γ Z Ω v γ − h ( x ) ϕ ε ( u ) + C. Since ϕ ε ( s ) ≤ ρ ( ρ − s for all s > , we finally deduce that Z { u>ε } ρ |∇ u | u ≤ C (cid:18)Z Ω v γ − | h ( x ) | u + 1 (cid:19) = C (cid:18)Z Ω | h ( x ) | v + 1 (cid:19) . Therefore, we let ε tend to zero and by virtue of Fatou’s lemma we obtain that Z Ω |∇ u | u ≤ C (cid:18)Z Ω | h ( x ) | v + 1 (cid:19) . Now, in [41] it is proven that R Ω |∇ u | u = + ∞ . Therefore, R Ω | h ( x ) | v = + ∞ and the proof isfinished. (cid:3) We are ready now to prove Theorem 1.1.
Proof of Theorem 1.1.
The existence of a solution u ∈ H (Ω) (not necessarily bounded) is aconsequence of [32, Theorem 3.1]. Moreover, since q > N , the well-known Stampacchia’s lemma(see [40]) implies that u ∈ L ∞ (Ω) .On the other hand, Lemma 5.4 leads to u ∈ C (Ω) . Furthermore, u ∈ W ,N loc (Ω) by virtue ofLemma 5.2 and Remark 5.3 in the Appendix below. Therefore, the uniqueness of solution followsdirectly from Theorem 2.3.Finally, it is clear that, if h (cid:13) has compact support, then R Ω h ( x ) u < + ∞ for every solution u to (1.3). Thus, Theorem 2.12 implies the nonexistence part of the theorem. (cid:3) Now we prove Theorem 2.4. Since the proofs of Theorem 2.6 and Theorem 2.7 are analogous,we will only make some comments below.
Proof of Theorem 2.4.
We divide the proof into several steps.
Step 1) An auxiliary problem
In addition to the hypotheses of Theorem 2.4, let us further assume for now that p < ∗ − and g (cid:13) . At the end of the proof of the theorem we will make the extension to general p, g .For every (cid:12) v ∈ C (Ω) , t ≥ , λ > , we consider the following problem:( Q t ) − ∆ u + ( v + δ ) g ( x, v ) |∇ u | u + δ = λf ( v ) + tv σ , x ∈ Ω ,u > , x ∈ Ω ,u = 0 , x ∈ ∂ Ω , where σ is the same real number that appears in condition ( g ∗ ); note that σ ∈ (0 , since g (cid:13) .It is clear that, if u is a solution to ( Q t ) with t = 0 and v = u , then it is actually a solution to( P λ ). In this step we will show that, if v (cid:13) , there exists a unique finite energy solution to ( Q t ).Observe that, if (cid:12) v ∈ C (Ω) , then (cid:12) λf ( v ) + tv σ ∈ C (Ω) too. Therefore, if δ = 0 , thenTheorem 1.1 implies directly that there exists a unique finite energy solution to ( Q t ), that wewill denote as u . On the other hand, let us assume that δ > . Then, u is clearly a subsolutionto ( Q t ). Let us consider the unique solution w ∈ H (Ω) ∩ C (Ω) to the linear problem − ∆ w = λf ( v ) + tv σ , x ∈ Ω ,w > , x ∈ Ω ,w = 0 , x ∈ ∂ Ω . Then, w is a supersolution to ( Q t ). Furthermore, u ≤ w in Ω by comparison, so [13, Théorème 3.1]implies that there exists a finite energy solution u δ to ( Q t ) such that < u ≤ u δ ≤ w in Ω .Moreover, every solution to ( Q t ) belongs to C (Ω) ∩ W ,N loc (Ω) by virtue of Lemma 5.4, Lemma 5.2and Remark 5.3. Also, the function ( x, s ) ( v ( x )+ δ ) g ( x,v ( x )) s + δ satisfies ( g ) and ( g ր ). Therefore,Theorem 2.3 implies that u δ is the unique solution to ( Q t ). Step 2) Existence of a finite energy solution to ( P λ ) if p < ∗ − and g (cid:13) . Let X = { w ∈ C (Ω) : w ( x ) ≥ ∀ x ∈ Ω } . We have shown in the previous step that theoperator K : X × [0 , + ∞ ) → X given by ( K ( v, t ) = u, ∀ v ∈ X \ { } , t ≥ , where u is the unique solution to ( Q t ) ,K (0 , t ) = 0 , ∀ t ≥ . is well-defined. We aim to prove that there exists (cid:12) u ∈ X such that K ( u,
0) = u , which isequivalent to finding a solution to ( P λ ). LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 25
We first prove that K is continuous. Indeed, let { v n } ⊂ X and { t n } ⊂ [0 , + ∞ ) be such that v n → v in C (Ω) for some v ∈ X and t n → t for some t ≥ . Let us denote u n = K ( v n , t n ) and h n = λf ( v n ) + t n v σn . Taking into account that g ≥ , it is clear that(4.6) − ∆ u n ≤ h n , x ∈ Ω , ∀ n. We know that there exists
C > such that h n ≤ C for all n . Then, (4.6) leads to − ∆ u n ≤ C in Ω .From this inequality, it is straightforward to prove that { u n } is bounded in H (Ω) . Hence, up toa subsequence, u n ⇀ u weakly in H (Ω) and u n → u strongly in L m (Ω) for some ≤ u ∈ H (Ω) and for all m ∈ [1 , ∗ ) . Furthermore, the inequality (4.6) yields the following estimate by virtueof the well-known Stampacchia’s lemma (see [40]):(4.7) k u n k L ∞ (Ω) ≤ C k h n k L ∞ (Ω) , ∀ n. In particular, if v ≡ , (4.7) yields directly that u n → strongly in C (Ω) . That is to say, K iscontinuous in { } × [0 , + ∞ ) .Let us assume on the contrary that v (cid:13) . In this case, we will prove following [32] that { u n } is locally bounded away from zero. Indeed, first of all observe that, using that h n ≤ C , wededuce from (4.7) that { u n } is bounded in L ∞ (Ω) . Hence, using ( f ∗ ), ( g ∗ ) and the L ∞ estimateon u n , one can easily check that the function w n = u − σn satisfies − ∆ w n ≥ (1 − σ ) λv pn u σn ≥ Cv pn , x ∈ Ω , for some constant C > . On the other hand, let z n ∈ H (Ω) ∩ C (Ω) be the unique solution tothe linear problem ( − ∆ z n = Cv pn , x ∈ Ω ,z n = 0 , x ∈ ∂ Ω . By comparison, w n ≥ z n for all n . Moreover, it is clear that { z n } is bounded in H (Ω) and,following [32] (which in turn is based on [28]), we also have that { z n } is bounded in C ,α (Ω) forsome α ∈ (0 , . Hence, passing to a subsequence, z n converges weakly in H (Ω) and strongly in C (Ω) to the unique solution z ∈ H (Ω) ∩ C (Ω) to ( − ∆ z = Cv p , x ∈ Ω ,z = 0 , x ∈ ∂ Ω . Since v (cid:13) , the strong maximum principle implies that, for all ω ⊂⊂ Ω , there exists c ω > suchthat z ≥ c ω in ω . Furthermore, since z n → z uniformly in Ω , it follows that z n ≥ z − c ω in Ω for all n large enough. In sum, u − σn = w n ≥ z n ≥ c ω , x ∈ ω for all n large enough, as we claimed.It is clear now that { ∆ u n } is bounded in L loc (Ω) . Hence, from [12] we deduce that ∇ u n → ∇ u a.e. in Ω . We may now pass to the limit using Fatou’s lemma twice, similarly as in the proof ofProposition 3.7 (Case 1, Step 1.3), so that u is the unique solution to ( Q t ) i.e. u = K ( v, t ) .It remains to prove that u n → u strongly in C (Ω) . Indeed, since { u n } is locally bounded awayfrom zero, Lemma 5.2 implies that, for every ω ⊂⊂ Ω , one may take a subsequence such that u n → u strongly in C ( ω ) . Actually, the uniqueness of u assures that the original sequence { u n } , and not merely a subsequence, converges itself to u strongly in C ( ω ) . On the other hand, let ζ n ∈ H (Ω) ∩ C (Ω) be the unique solution to ( − ∆ ζ n = h n , x ∈ Ω ,ζ n = 0 , x ∈ ∂ Ω . By comparison, u n ≤ ζ n in Ω for all n . Moreover, ζ n → ζ strongly in C (Ω) , where ζ ∈ H (Ω) ∩ C (Ω) is the unique solution to ( − ∆ ζ = λf ( v ) + tv σ , x ∈ Ω ,ζ = 0 , x ∈ ∂ Ω . In particular, for fixed ε > , we deduce that | u n ( x ) − u ( x ) | ≤ ζ n ( x ) + u ( x ) < ζ ( x ) + u ( x ) + ε ∀ x ∈ Ω , ∀ n large enough.Therefore, since u, ζ ∈ C (Ω) with u = ζ = 0 on ∂ Ω , there exits ω ⊂⊂ Ω such that | u n ( x ) − u ( x ) | < ε ∀ x ∈ Ω \ ω, ∀ n large enough.Since u n → u strongly in C ( ω ) , we conclude that | u n ( x ) − u ( x ) | < ε ∀ x ∈ Ω , ∀ n large enough.In other words, u n → u strongly in C (Ω) , so K is continuous.We prove now that K is compact, i.e. it maps bounded sets to relatively compact sets.Indeed, let { v n } ⊂ X and { t n } ⊂ [0 , + ∞ ) be bounded sequences. Taking subsequences, v n → v in the weak- ⋆ topology of L ∞ (Ω) for some v ∈ L ∞ (Ω) , while t n → t for some t ≥ . This isenough to pass to the limit in the equations as above. We conclude that, up to subsequences, K ( v n , t n ) → K ( v, t ) strongly in C (Ω) . This proves that K is compact.We will prove next that there exist < r < R and t ≥ such that(1) u = sK ( u, ∀ s ∈ [0 , , ∀ u ∈ X with k u k L ∞ (Ω) = r ,(2) u = K ( u, t ) ∀ t ≥ , ∀ u ∈ X with k u k L ∞ (Ω) = R ,(3) u = K ( u, t ) ∀ t ≥ t , ∀ u ∈ X with k u k L ∞ (Ω) ≤ R .In order to prove item (1), let us assume by contradiction that, for all r > , there exist s ∈ [0 , and u ∈ X with k u k L ∞ (Ω) = r such that u = sK ( u, . In particular, u > in Ω and ( − ∆ u ≤ sλf ( u ) , x ∈ Ω ,u = 0 , x ∈ ∂ Ω . Since ( f ) holds, we can choose r > such that λf ( t ) ≤ λ t for all t ∈ [0 , r ] , where λ standsfor the principal eigenvalue of − ∆ in Ω with zero Dirichlet boundary conditions. Furthermore, u ≤ r in Ω , so we have that − ∆ u ≤ λ u, x ∈ Ω ,u > , x ∈ Ω ,u = 0 , x ∈ ∂ Ω . This is a clear contradiction with the definition of λ .On the other hand, if we take R > C , where k u k L ∞ (Ω) ≤ C (see Proposition 3.7) then it isclear that (2) holds. Moreover, if we take t > t , where t > is given by Proposition 3.5, then(3) also holds. LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 27
In conclusion, [21, Proposition 2.1 and Remark 2.1] can be applied and in consequence weobtain a positive fixed point of K , i.e. a finite energy solution to ( P λ ). Step 3) Extension to the general case
From this point we assume only the hypotheses of Theorem 2.4, i.e. g need not be non-negativeand p might be greater than or equal to ∗ − .Fix λ > . For some γ > that will be chosen later, let us consider the following problem:(4.8) − ∆ v + g γ ( x, v ) |∇ v | = λf γ ( v ) , x ∈ Ω ,v > , x ∈ Ω ,v = 0 , x ∈ ∂ Ω , where f γ ( s ) = b ( s + δ γ ) γ − γ f (cid:16) ( s + δ γ ) γ − δ (cid:17) , ∀ s ≥ ,g γ ( x, s ) = ( s + δ γ ) γ g (cid:16) x, ( s + δ γ ) γ − δ (cid:17) + γ − γ ( s + δ γ ) , a.e. x ∈ Ω , ∀ s > , and b > is a constant, which depends only on p, δ, γ , that will be chosen below too. It is easyto see that, if v is a solution to (4.8), then u = ( v + δ γ ) γ − δ is a solution to ( P λ ). Moreover, if v ∈ H (Ω) and δ > , then u ∈ H (Ω) too, so it has finite energy. We will now choose γ > sothat problem (4.8) admits a finite energy solution.Indeed, let us first denote p γ = γ − pγ . Obviously, p γ > . It is easy to see that the function s s p γ ( s + δ γ ) γ − γ h ( s + δ γ ) γ − δ i p is bounded in [0 , + ∞ ) . Then, using that f satisfies ( f ∗ ), b > can be chosen (depending onlyon p, δ, γ ) such that f γ ( s ) ≥ s p γ ∀ s ≥ . Moreover, since f satisfies ( f ∗ ), it is clear that f γ ( s ) ≤ abγ ( s + δ γ ) p γ ∀ s ≥ . We have proved that f γ satisfies ( f ∗ ). Taking into account that f satisfies ( f ∞ ) and ( f ), it isstraightforward to check that also f γ satisfies ( f ∞ ) and ( f ) respectively.On the other hand, since g satisfies ( g ∞ ), then lim s → + ∞ (cid:13)(cid:13)(cid:13)(cid:13) sg γ ( · , s ) − µ + γ − γ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (Ω) = 0 . Using that max x ∈ Ω µ ( x ) < ∗ − − p ∗ − , it is easy to see that max x ∈ Ω µ ( x ) + γ − γ < ∗ − − p γ ∗ − . Thus, g γ satisfies ( g ∞ ). It is also immediate to check that g γ satisfies ( g ) using that g does too.Notice finally that, since g satisfies ( g ∗ ), then τ + γ − γ ≤ ( s + δ γ ) g γ ( s, x ) ≤ σ + γ − γ a.e. x ∈ Ω , ∀ s > . It is clear that σ + γ − γ − < τ + γ − γ ≤ σ + γ − γ < . Therefore, g γ satisfies ( g ∗ ).If we now choose γ > large enough so that τ + γ − γ > and p γ < ∗ − , then we mayapply the previous step in order to obtain a finite energy solution to (4.8). The proof is nowfinished. (cid:3) Proofs of Theorem 2.6 and Theorem 2.7.
The proofs are analogous to that for Theorem 2.4. Theonly essential difference lies in how the estimates on the solutions are obtained. In fact, insteadof Proposition 3.7, we have to apply Proposition 3.9 for Theorem 2.6 and both Proposition 3.8and Proposition 3.10 for Theorem 2.7. (cid:3)
Here we carry out the proof of Theorem 2.8.
Proof of Theorem 2.8.
Let us define the Carathéodory function ¯ g : Ω × (0 , + ∞ ) → R by ¯ g ( x, s ) = g ( x, s ) if x ∈ Ω , s ∈ (0 , s ) ,s g ( x, s ) s if x ∈ Ω , s ≥ s . Let us also consider the continuous function ¯ f : [0 , + ∞ ) → [0 , + ∞ ) defined by ¯ f ( s ) = f ( s ) if s ∈ (0 , s ) ,f ( s ) s p s p if s ≥ s . It is clear that ¯ f and ¯ g satisfy the hypotheses of Theorem 2.7. Thus, if we write ( ¯ P λ ) fordenoting problem ( P λ ) with ¯ g and ¯ f instead of g and f , then Theorem 2.7 implies that thereexists a solution u λ to ( ¯ P λ ) for all λ > that satisfies λ p − k u λ k L ∞ (Ω) ≤ C ∀ λ > . In particular, there exists λ > such that k u λ k L ∞ (Ω) < s for every λ ≥ λ . Hence, u λ is, infact, a solution to ( P λ ) for every λ ≥ λ . The proof is finished. (cid:3) It is left to prove Theorem 2.10.
Proof of Theorem 2.10.
Let σ ∈ (cid:16) , min n , ∗ − − p ∗ − o(cid:17) . By virtue of Remark 2.9, we may take δ ∈ (0 , such that sg ( x, s ) ≤ σ ∀ ( x, s ) ∈ Ω × (0 , δ ] . Notice that δ can be chosen as small as necessary. In fact, it will be taken small enough severaltimes throughout the proof, depending its size only on N, p, G .Let us define the function ¯ g : Ω × (0 , + ∞ ) → [0 , + ∞ ) by ¯ g ( x, s ) = g ( x, s ) if ( x, s ) ∈ Ω × (0 , δ ] ,δg ( x, δ ) s if ( x, s ) ∈ Ω × ( δ, + ∞ ) . It is clear that ¯ g satisfies ( g ∗ ), ( g ∞ ) and ( g ). Thus, if we write ( ¯ P λ ) for denoting problem( P λ ) with ¯ g instead of g , then Theorem 2.4 implies that there exists a solution to ( ¯ P λ ) for all LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 29 λ > . We aim now to prove that the hypotheses of Proposition 3.11 hold true, so that thereexists C > such that(4.9) λ p − k u k L ∞ (Ω) ≤ C ∀ u solution to ( ¯ P λ ) with λ > . Observe that, if this is true, in particular there exists λ > such that k u k L ∞ (Ω) ≤ δ for everysolution u to ( ¯ P λ ) with λ ≥ λ . Hence, the solutions to ( ¯ P λ ) with λ ≥ λ are, in fact, solutionto ( P λ ). Thus, the proof will finish as soon as we prove (4.9).We will now prove that (3.11) in Proposition 3.11 holds. Indeed, for any L > , x ∈ Ω , letus consider the function ψ : [0 , + ∞ ) → [0 , + ∞ ) given by ψ ( s ) = Z s e − L R tδL ¯ g ( x , rL ) dr dt ∀ s ≥ . It is straightforward to check that ψ ( s ) = L Z sL e − R tδ ¯ g ( x ,r ) dr dt ∀ s ≥ . We will prove next that s ψ ′ ( s ) s p ψ ( s ) ∗− is decreasing for all s > , or equivalently, that(4.10) ( p − s ¯ g ( x , s )) ψ ( sL ) L < (2 ∗ − sψ ′ ( sL ) ∀ s > . Notice that neither ψ ( sL ) L nor ψ ′ ( sL ) depends on L .In order to prove (4.10), we distinguish two possibilities: either < s ≤ δ or s > δ . Assumefirst that < s ≤ δ . Then, ( p − s ¯ g ( x , s )) ψ ( sL ) L − (2 ∗ − sψ ′ ( sL )=( p − sg ( x , s )) Z s e − R tδ g ( x ,r ) dr dt − (2 ∗ − se − R sδ g ( x ,r ) dr . It follows from the mean value theorem that there exists θ ∈ (0 , such that ( p − sg ( x , s )) Z s e − R tδ g ( x ,r ) dr dt − (2 ∗ − se − R sδ g ( x ,r ) dr = se R δs g ( x ,r ) dr h ( p − sg ( x , s )) e R sθs g ( x ,r ) dr − (2 ∗ − i . Thanks to (G) we derive ( p − sg ( x , s )) e R sθs g ( x ,r ) dr − (2 ∗ − ≤ pe R δ G ( r ) dr − (2 ∗ − . Using that lim δ → R δ G ( r ) dr = 0 and p < ∗ − , we are allowed to choose δ small enough sothat pe R δ G ( r ) dr < ∗ − . Hence, (4.10) holds for < s ≤ δ .We assume now that s > δ . In this case, ψ ( s ) = AL + δ τ L (cid:0) sL (cid:1) − τ − δ − τ − τ , where τ = δg ( x , δ ) and A = ψ ( δL ) L = Z δ e R δt g ( x ,r ) dr dt. Recall that τ ≤ σ < ∗ − − p ∗ − . Then, p − τ − τ ≤ p − σ − σ < ∗ − . Let us denote γ = 2 ∗ − − p − σ − σ . We deduce that ( p − s ¯ g ( x , s )) ψ ( sL ) L − (2 ∗ − sψ ′ ( sL )= ( p − τ ) (cid:18) A + δ τ s − τ − δ − τ − τ (cid:19) − (2 ∗ − δ τ s − τ ≤ (2 ∗ − (cid:0) (1 − τ ) A + δ τ ( s − τ − δ − τ ) − δ τ s − τ (cid:1) − γ (cid:18) A + δ τ s − τ − δ − τ − τ (cid:19) ≤ (2 ∗ − A − δ ) − γA = A (cid:18) (2 ∗ − (cid:18) − δA (cid:19) − γ (cid:19) ≤ A (2 ∗ − − δ Z δ e R δt G ( r ) dr dt ! − − γ . It is clear that lim δ → R δ e R δt G ( r ) dr dtδ = lim δ → e R δ G ( r ) dr R δ e − R t G ( r ) dr dtδ = lim δ → e − R δ G ( r ) dr = 1 . Then, we can take δ small enough so that (2 ∗ − − δ Z δ e R δt G ( r ) dr dt ! − < γ. In conclusion, (4.10) holds for s > δ , and thus, for all s > .In sum, it follows that the hypotheses of Proposition 3.11 are fulfilled for problem ( ¯ P λ ) so ourclaim was true. The proof is now finished. (cid:3) We conclude the section by proving Theorem 1.2, Theorem 1.3 and Theorem 1.4 as conse-quences of the general results in Section 2.
Proof of Theorem 1.2.
The first item of the theorem is a direct consequence of Theorem 2.4,while the second one follows from Theorem 2.12. (cid:3)
Proof of Theorem 1.3.
Let us denote g ( x, s ) = µ ( x )( s + δ ) γ . In order to prove the first item, we assumethat δ > and that µ satisfies (1.4). It is easy to see that inf ( x,s ) ∈ Ω × [0 , + ∞ ) ( s + δ ) g ( x, s ) = − k µ − k L ∞ (Ω) δ γ − , sup ( x,s ) ∈ Ω × [0 , + ∞ ) ( s + δ ) g ( x, s ) = k µ + k L ∞ (Ω) δ γ − . Therefore, τ ≤ ( s + δ ) g ( x, s ) ≤ σ for τ = − k µ − k L ∞ (Ω) δ γ − and σ = k µ + k L ∞ (Ω) δ γ − . Hence, (1.4) impliesthat g satisfies ( g ∗ ). On the other hand, it is clear that k ( s + δ ) g ( · , s ) k L ∞ (Ω) → as s → + ∞ . LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 31
Thus, g also satisfies ( g ∞ ). In conclusion, Theorem 2.4 implies that there exists a finite energysolution to (1.1) for any λ > .On the contrary, let us assume now that δ = 0 and µ ( x ) ≥ τ > for a.e. x ∈ Ω \ ω . Then,( g ) holds. Thus, Theorem 2.12 implies that problem (1.1) admits no solution for any λ > . (cid:3) Proof of Theorem 1.4.
Let us denote g ( x, s ) = µ ( x )( s + δ ) γ . We know that, for every ε > , thereexists s > such that(4.11) s | g ( x, s ) | ≤ k µ k L ∞ (Ω) s ( s + δ ) γ < ε ∀ s ∈ (0 , s ) , a.e. x ∈ Ω . Therefore, ( e g ) holds (in particular, lim s → sg ( x, s ) obviously exists for a.e. x ∈ Ω ). If we assumethat (cid:12) µ ∈ C (Ω) and p < ∗ − , then the hypotheses of Theorem 2.10 are fulfilled, so weconclude.Let us assume on the contrary that p < N +1 N − . Then, in (4.11) we may take σ = − τ = ε ∈ (cid:16) , min n , N +1 − ( N − p o(cid:17) in such a way that g satisfies ( e g ∗ ). Thus, the result follows afterapplying Theorem 2.8.The proof is finished. (cid:3) Appendix
Technical lemma.
In this subsection we prove a technical result that is required by theblow-up method.
Lemma 5.1.
Let Ω ⊂ R N ( N ≥ be a bounded domain with boundary of class C , λ > , g : Ω × (0 , + ∞ ) → R be a Carathéodory function, and f : [0 , + ∞ ) → R be a continuous function.Then, for every x ∈ ∂ Ω , there exist U ⊂ R N a neighborhood of x and an injective and C map y : U → R N , with C inverse, such that V = y ( U ∩ Ω) ⊂ R N + , Γ = y ( U ∩ ∂ Ω) ⊂ ∂V ∩ ∂ R N + and,if u is a solution to ( P λ ) , then the function v = u ◦ y − : V → (0 , + ∞ ) is a solution to (5.1) − div ( M ( y ) ∇ v ) + b ( y ) ∇ v + j ( y, v ) M ( y ) ∇ v ∇ v = f ( v ) , y ∈ V,v > , y ∈ V,v = 0 , y ∈ Γ , where M ∈ C ( V ) N × N is uniformly elliptic, b ∈ C ( V ) N and j ( · , · ) = g ( y − ( · ) , · ) , being M and b independent of u .Proof. Since ∂ Ω is of class C , there exist U ⊂ R N a neighborhood of x and a C function ψ : U ′ → R , where U ′ = { x ′ ∈ R N − : ∃ x N ∈ R , ( x ′ , x N ) ∈ U } , such that ψ ( x ′ ) < x N ∀ ( x ′ , x N ) ∈ U ∩ Ω ,ψ ( x ′ ) = x N ∀ ( x ′ , x N ) ∈ U ∩ ∂ Ω . Let us define the change of variables y : U → R N by y ( x ) = ( x ′ , x N − ψ ( x ′ )) ∀ x ∈ U. It is clear that y ( x ) ∈ R N + ∀ ( x ′ , x N ) ∈ U ∩ Ω ,y ( x ) ∈ ∂ R N + ∀ ( x ′ , x N ) ∈ U ∩ ∂ Ω . This proves that V ⊂ R N + and Γ ⊂ ∂ R N + . Moreover, it is a simple exercise to prove that Γ ⊂ ∂V .It is also easy to see that the function y − : y ( U ) → U given by y − ( z ) = ( z ′ , z N + ψ ( z ′ )) ∀ z ∈ y ( U ) is the inverse function of y . Note that y − is well defined since z ′ ∈ U ′ for every z ∈ y ( U ) .Let us now define v : V ∪ Γ → R by v ( z ) = u ( y − ( z )) ∀ z ∈ V ∪ Γ . Observe that v = 0 on Γ and u ( x ) = v ( y ( x )) for all x ∈ U ∩ Ω . We will show next that v satisfiesan equation in V .Now we compute the derivatives that we need. We emphasize that such derivatives can beunderstood in a pointwise sense due to Remark 5.3 and to the C regularity of ψ . We stress alsothat, as ψ does not depend on x N , it will be understood that ∂ψ∂x N ( x ) = 0 for x ∈ U .a) Dy ( x ) = · · · ... ...... . . . ... · · · − ∂ψ∂x ( x ′ ) · · · · · · − ∂ψ∂x N − ( x ′ ) 1 , b) ∂u∂x i ( x ) = ∂v∂z i ( y ( x )) − ∂v∂z N ( y ( x )) ∂ψ∂x i ( x ′ ) , i = 1 , · · · , N − ,∂u∂x N ( x ) = ∂v∂z N ( y ( x )) , c) |∇ u ( x ) | = |∇ v ( y ( x )) | + (cid:16) ∂v∂z N ( y ( x )) (cid:17) |∇ ψ ( x ′ ) | − ∂v∂z N ( y ( x )) ∇ v ( y ( x )) ∇ ψ ( x ′ ) , d) ∂ u∂x i ( x ) = ∂ v∂z ∂z i ( y ( x )) ∂y ∂x i ( x ) + · · · + ∂ v∂z N ∂z i ( y ( x )) ∂y N ∂x i ( x ) − ∂v∂z N ( y ( x )) ∂ ψ∂x i ( x ′ ) − (cid:20) ∂ v∂z ∂z N ( y ( x )) ∂y ∂x i ( x ) + · · · + ∂ v∂z N ( y ( x )) ∂y N ∂x i ( x ) (cid:21) ∂ψ∂x i ( x ′ )= ∂ v∂z i ( y ( x )) − ∂ v∂z i ∂z N ( y ( x )) ∂ψ∂x i ( x ′ ) + ∂ v∂z N ( y ( x )) (cid:18) ∂ψ∂x i ( x ′ ) (cid:19) − ∂v∂z N ( y ( x )) ∂ ψ∂x i ( x ′ ) , i = 1 , · · · , N − , e) ∆ u = ∆ v − ∇ ∂v∂z N ∇ ψ + ∂ v∂z N |∇ ψ | − ∂v∂z N ∆ ψ. LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 33
Let us denote j ( z, s ) = g ( y − ( z ) , s ) for a.e. z ∈ V and for all s > . Thus, v = v ( y ) (fromthis point y will simply denote variable in V ) satisfies the equation − ∆ v − ∂ v∂y N |∇ ψ | + 2 ∇ ∂v∂y N ∇ ψ + ∂v∂y N ∆ ψ = f ( v ) − j ( y, v ) |∇ v | + (cid:18) ∂v∂y N (cid:19) |∇ ψ | − ∂v∂y N ∇ v ∇ ψ ! , y ∈ V. (5.2)Let us define the matrix M ( y ) = · · · − ∂ψ∂x ( y ′ )0 1 ... ...... . . . ... · · · − ∂ψ∂x N − ( y ′ )0 · · · · · · |∇ ψ ( y ′ ) | , y ∈ V, and also the vector b ( y ) = (0 , · · · , , − ∆ ψ ( y ′ )) , y ∈ V . Then, one can check that v is a solutionto (5.1).Moreover, let a > be such that ( a − k∇ ψ k L ∞ ( V ) < . Then, by Cauchy-Schwarz’s andYoung’s inequalities, the following holds for every ξ ∈ R N : M ( y ) ξξ = | ξ | − ξ N ∇ ψξ ′ + ξ N |∇ ψ | ≥ | ξ | − ( a − ξ N |∇ ψ | − a | ξ ′ | = (cid:18) − a (cid:19) | ξ ′ | + (1 − ( a − |∇ ψ | ) ξ N ≥ (cid:18) − a (cid:19) | ξ ′ | + (1 − ( a − k∇ ψ k L ∞ ( V ) ) ξ N ≥ min (cid:26) − a , − ( a − k∇ ψ k L ∞ ( V ) (cid:27) | ξ | . Then, M is uniformly elliptic. The proof is finished. (cid:3) Local Hölder estimate and global continuity.
This subsection is devoted to proving a local Hölder condition on the solutions to the problemsthat concern this manuscript. In particular, the solutions are continuous in the interior of Ω . Wewill show at the end of the section that the solutions are, in fact, continuous up to the boundary. Lemma 5.2.
Let g : Ω × (0 , + ∞ ) → R be a Carathéodory function satisfying (2.1) and let h ∈ L q (Ω) for some q > N . Then, for every M > c > and every ω ⊂⊂ Ω , there exist α ∈ (0 , , L > such that every solution u ∈ H loc (Ω) ∩ L ∞ (Ω) to (5.3) ( − ∆ u + g ( x, u ) |∇ u | = h ( x ) , x ∈ Ω ,c ≤ u ≤ M, x ∈ Ω , satisfies (5.4) k u k C ,α ( ω ) ≤ L. Proof.
Let us consider m = max s ∈ [ c,M ] g ( s ) , where g : (0 , + ∞ ) → [0 , + ∞ ) is the continuousfunction given by (2.1), and let us take δ ∈ (cid:16) , m (cid:17) . With the notation of [28, Chapter 2,Section 6, p. 81], we will show that u belongs to the class B (cid:16) Ω , M, γ, δ, q (cid:17) for some γ > tobe determined later. This implies (5.4) by virtue of [28, Theorem 6.1, p.90].Indeed, let us fix ρ > and let B ρ be an open ball of radius ρ such that B ρ ⊂ Ω . Let us fixalso σ ∈ (0 , , and consider a function ζ ∈ C ∞ c ( B ρ ) satisfying that ≤ ζ ≤ in B ρ , ζ ≡ in theconcentric open ball B ρ − σρ , and |∇ ζ | < bσρ in B ρ for some constant b > independent of ρ, σ .Let us denote k ρ = sup x ∈ B ρ u ( x ) − δ , and take k ≥ k ρ . We also consider the set A k,ρ = { x ∈ B ρ : u ( x ) ≥ k } . Clearly, ( u − k ) + ζ ∈ H (Ω) ∩ L ∞ (Ω) and has compact support in Ω , so it may be taken as testfunction in (5.3). Thus we obtain Z A k,ρ ζ ( u − k ) ∇ ζ ∇ u + Z A k,ρ |∇ u | ζ + Z A k,ρ g ( x, u ) |∇ u | ( u − k ) ζ = Z A k,ρ h ( x )( u − k ) ζ . On the one hand, by (2.1) and by the definitions of k ρ , m we deduce that − Z A k,ρ g ( x, u ) |∇ u | ( u − k ) ζ ≤ δm Z A k,ρ |∇ u | ζ . On the other hand, Young’s inequality yields − Z A k,ρ ζ ( u − k ) ∇ ζ ∇ u ≤ Z A k,ρ ( u − k ) |∇ ζ | + 12 Z A k,ρ |∇ u | ζ . Therefore, we derive (cid:18) − δm (cid:19) Z A k,ρ |∇ u | ζ ≤ Z A k,ρ ( u − k ) |∇ ζ | + Z A k,ρ h ( x )( u − k ) ζ . Next, it follows from Hölder’s inequality that Z A k,ρ h ( x )( u − k ) ζ ≤ δ k h k L q (Ω) | A k,ρ | q ′ and Z A k,ρ ( u − k ) ζ ≤ bρ σ Z A k,ρ ( u − k ) ≤ bρ σ k ( u − k ) k L ∞ ( A k,ρ ) | A k,ρ |≤ bC ( N ) ρ − Nq σ k ( u − k ) k L ∞ ( A k,ρ ) | A k,ρ | q ′ , where | B ρ | = C ( N ) q ρ N . In sum,(5.5) Z A k,ρ − σρ |∇ u | ≤ γ ρ − Nq σ k ( u − k ) k L ∞ ( A k,ρ ) + 1 ! | A k,ρ | q ′ , where γ = max (cid:8) bC ( N ) , δ k h k L q (Ω) (cid:9) − δm . Furthermore, notice that v = − u satisfies − ∆ v − g ( x, − v ) |∇ v | = − h ( x ) , x ∈ Ω . LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 35
Therefore, it is straightforward to check that the inequality (5.5) also holds by substituting u with − u . In conclusion, u ∈ B (cid:16) Ω , M, γ, δ, q (cid:17) and the proof is finished. (cid:3) Remark 5.3.
Let g : Ω × (0 , + ∞ ) → R be a Carathéodory function satisfying (2.1), h ∈ L q (Ω) for some q > N and u ∈ H loc (Ω) ∩ L ∞ (Ω) . Using the local Hölder regularity given by Lemma 5.2one can prove that, if u ∈ H loc (Ω) ∩ L ∞ (Ω) is a solution to (5.6) that is locally bounded awayfrom zero, then u ∈ W , q loc (Ω) . The proof is based on a standard bootstrap argument, see [15,Appendix] for further details. Moreover, combining this local summability of the gradients withthe classical Calderon-Zygmund regularity theory one can easily prove that, if h ∈ L ∞ (Ω) , then u ∈ W ,q loc (Ω) for every q < ∞ .We will show next that every solution to ( H ) is continuous up to ∂ Ω . Lemma 5.4.
Let g : Ω × (0 , + ∞ ) → [0 , + ∞ ) be a Carathédory function satisfying (2.1) and let h ∈ L q (Ω) for some q > N . Then, every solution to ( H ) belongs to C (Ω) .Proof. Let u ∈ H loc (Ω) ∩ L ∞ (Ω) be a solution to ( H ). Clearly, Lemma 5.2 implies that u ∈ C (Ω) .It is left to prove the continuity on the boundary. In order to do so, recall that there exists γ > such that u γ ∈ H (Ω) . We may assume without loss of generality that γ > . Let us denote v = u γ . Using that g ≥ , we deduce that − ∆ v = − γ ( γ − u γ − |∇ u | + γu γ − ( − ∆ u ) ≤ γu γ − h ( x ) ≤ C | h ( x ) | , x ∈ Ω , for some constant C > . On the other hand, let us consider the unique solution w ∈ H (Ω) ∩ C (Ω) to the following problem: ( − ∆ w = | h ( x ) | , x ∈ Ω ,w = 0 , x ∈ ∂ Ω . By standard comparison, < v ≤ w in Ω . Therefore, lim x → y v ( x ) = 0 for every y ∈ ∂ Ω and, inparticular, u ∈ C (Ω) . (cid:3) Global Hölder estimates.
In the previous subsection we have shown that the solutions to the problems in this paperare locally Hölder continuous. In the proof, we have strongly used the fact that the solutionsare locally bounded away from zero in order to avoid the possible singularity of g ( x, u ( x )) as x approaches ∂ Ω . In the present subsection we aim to prove a global Hölder estimate, i.e. thesingularity cannot be avoided since we need a Hölder estimate to hold even near ∂ Ω . Therefore,we perform a different proof that requires g to satisfy the stronger condition ( g ∗ ). Lemma 5.5.
Let Ω ⊂ R N be a smooth bounded domain. Let { M n } ⊂ L ∞ (Ω) N × N and { b n } ⊂ L ∞ (Ω) N be bounded sequences in the norm of their respective spaces. Assume that there exists η > such that M n ( y ) ξξ ≥ η | ξ | a.e. y ∈ Ω , ∀ ξ ∈ R N , ∀ n. For some p > , δ, τ ≥ , σ ∈ (0 , independent of n , let f : [0 , + ∞ ) → [0 , + ∞ ) be a continuousfunction satisfying ( f ∗ ) and, for all n , let g n : Ω × (0 , + ∞ ) → [0 , + ∞ ) be a Carathédory functionsatisfying ( g ∗ ) . Let { v n } ⊂ H (Ω) ∩ C (Ω) be such that < v n ≤ C in Ω for all n and for some C > . For some sequences { L n } ⊂ (0 , + ∞ ) , { ǫ n } ⊂ [0 , , assume that v n satisfies − div ( M n ( y ) ∇ v n )+ b n ( y ) ∇ v n + 1 L n g n (cid:18) y, v n L n (cid:19) M n ( y ) ∇ v n ∇ v n = L pn f (cid:18) v n L n (cid:19) + ǫ n v σn , y ∈ Ω , ∀ n. If δ = 0 , assume in addition that there exists γ ∈ (cid:0) − τ , − σ (cid:3) such that v γn ∈ H (Ω) for all n .For some (possibly empty) set Γ ⊂ ∂ Ω that is open in the topology of ∂ Ω and connected, let ussuppose also that v n = 0 , y ∈ Γ . Furthermore, if { L n } diverges, assume additionally that δ = 0 and ǫ n = 0 for all n . Then, forevery smooth bounded domain ω satisfying that ω ⊆ Ω ∪ Γ , there exist α ∈ (0 , , C > suchthat, taking a (not relabeled) subsequence if necessary, k v n k C ,α ( ω ) ≤ C ∀ n. Proof.
Let us denote δ n = L n δ and consider the function u n = ( v n + δ n ) γ − δ γn for γ ∈ (cid:0) − τ , − σ (cid:3) ;if δ = 0 , then γ must be chosen so that u n ∈ H (Ω) for all n . It is easy to see that u n ∈ C (Ω) , < u n ≤ C in Ω , u n = 0 on Γ , and it satisfies(5.6) − div ( M n ( y ) ∇ u n ) + b n ( y ) ∇ u n = ˜ g n ( y, u n ) M n ( y ) ∇ u n ∇ u n + ˜ f n ( u n ) , y ∈ Ω , where ˜ g n ( y, s ) = 1 − γ − ( s + δ γn ) γ L n g n (cid:18) y, ( s + δ γn ) γ L n − δ (cid:19) γ ( s + δ γn ) a.e. y ∈ Ω , ∀ s > , ∀ n, ˜ f n ( s ) = γ ( s + δ γn ) γ − γ L pn f ( s + δ γn ) γ L n − δ ! + ǫ n (cid:16) ( s + δ γn ) γ − δ n (cid:17) σ ! a.e. y ∈ Ω , ∀ n. We claim that { u n ˜ f n ( u n ) } admits a subsequence bounded in L ∞ (Ω) and, furthermore,(5.7) ≤ s ˜ g n ( y, s ) ≤ c a.e. y ∈ Ω , ∀ s > , ∀ n, for some c ∈ (0 , . In order to prove the claim, we first observe that ( f ∗ ) implies ˜ f n ( s ) ≤ a ( s + δ γn ) p − γγ + ǫ n ( s + δ γn ) σ − γγ . On the one hand, let us assume that { L n } is not divergent. Then, it admits a (not relabeled)bounded subsequence. Hence, taking into account that p > , it clearly holds that au n ( u n + δ γn ) p − γγ ≤ C ∀ n. Besides, since γ ≥ − σ , then σ − γ ≥ , so γǫ n u n ( u n + δ γn ) σ − γγ ≤ ( u n + δ γn ) σ − γγ ≤ C ∀ n. Thus, { u n ˜ f n ( u n ) } is bounded in L ∞ (Ω) . On the other hand, assume that { L n } diverges. Then, δ n = ǫ n = 0 for all n , so ˜ f n ( u n ) ≤ au p − γγ n ≤ C for all n and, again, { u n ˜ f n ( u n ) } is bounded in L ∞ (Ω) .Let us now verify that (5.7) holds. Indeed, from ( g ∗ ) and γ ∈ (cid:0) − τ , − σ (cid:3) , it follows ≤ − γ − σγ ≤ ( s + δ γn )˜ g n ( y, s ) ≤ − γ − τγ < . Thus, our claim holds.
LOW-UP FOR SINGULAR PROBLEMS WITH NATURAL GROWTH 37
In sum, we may apply the arguments in [15, Appendix] without relevant changes to prove that k u n k C ,α ( ω ) ≤ C for some C > , α ∈ (0 , . In particular, using that the function s s γ islocally Lipschitz for s ≥ , we have that | v n ( x ) − v n ( y ) | = | v n ( x ) + δ n − v n ( y ) − δ n |≤ C | ( v n ( x ) + δ n ) γ − ( v n ( y ) + δ n ) γ | = C | u n ( x ) − u n ( y ) | ≤ C | x − y | α ∀ x, y ∈ ω. In conclusion, k v n k C ,α ( ω ) ≤ C , as we wanted to prove. (cid:3) References [1] A. Ambrosetti, P.H. Rabinowitz,
Dual variational methods in critical point theory and applications.
J. Func-tional Analysis (1973), 349–381.[2] D. Arcoya, J. Carmona, T. Leonori, P.J. Martínez-Aparicio, L. Orsina, F. Pettita, Existence and nonexistenceof solutions for singular quadratic quasilinear equations . J. Differential Equations , (2009), 4006–4042.[3] D. Arcoya, J. Carmona, P.J. Martínez-Aparicio, Bifurcation for quasilinear elliptic singular BVP.
Comm.Partial Differential Equations (2011), no. 4, 670–692.[4] D. Arcoya, J. Carmona, P. J. Martínez-Aparicio, Comparison principle for elliptic equations in divergencewith singular lower order terms having natural growth.
Commun. Contemp. Math., (2017), 1650013, 11pp.[5] D. Arcoya, C. De Coster, L. Jeanjean, K. Tanaka, Remarks on the uniqueness for quasilinear elliptic equationswith quadratic growth conditions.
J. Math. Anal. Appl. (2014), 772–780.[6] D. Arcoya, L. Moreno-Mérida,
The effect of a singular term in a quadratic quasi-linear problem.
J. FixedPoint Theory Appl. (2017), 815–831.[7] D. Arcoya, J.L. Gámez, L. Orsina, I. Peral, Local existence results for sub-super-critical elliptic problems.
Commun. Appl. Anal. (2001), no. 4, 557–569.[8] D. Arcoya, S. Segura de León, Uniqueness of solutions for some elliptic equations with a quadratic gradientterm . ESAIM Control Optim. Calc. Var., (2010), 327–336.[9] G. Bianchi, Non–existence of positive solutions to semilinear elliptic equations on R N or R N + through themethod of moving planes. Comm. Partial Differential Equations (1997), no. 9–10, 1671–1690.[10] I. Birindelli, E. Mitidieri, Liouville theorems for elliptic inequalities and applications.
Proc. Roy. Soc. Edin-burgh Sect. A (1998), no. 6, 1217–1247.[11] L. Boccardo,
Dirichlet problems with singular and gradient quadratic lower order terms.
ESAIM ControlOptim. Calc. Var. (2008), 411–426.[12] L. Boccardo, F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolicequations.
Nonlinear Anal. (1992), no. 6, 581–597.[13] L. Boccardo, F. Murat, J.-P. Puel, Quelques propriétés des opérateurs elliptiques quasi linéaires . C. R. Acad.Sci. Paris Sér. I Math. (1988), 749–752.[14] L. Boccardo, L. Orsina, M. Porzio,
Existence results for quasilinear elliptic and parabolic problems withquadratic gradient terms and sources.
Adv. Calc. Var. (2011), no. 4, 3974–19.[15] J. Carmona, T. Leonori, S. López-Martínez, P.J. Martínez-Aparicio, Quasilinear elliptic problems with sin-gular and homogeneous lower order terms . Nonlinear Anal. (2019), 105-130.[16] J. Carmona, T. Leonori,
A uniqueness result for a singular elliptic equation with gradient term , Proc. Roy.Soc. Edinburgh Sect. A (2018), no. 5, 983–994.[17] J. Carmona, S. López-Martínez, P.J. Martínez-Aparicio,
Singular quasilinear elliptic problems with changingsign datum: existence and homogenization.
Rev. Mat. Complut. (2020), no. 1, 39–62.[18] J. Carmona, P.J. Martínez-Aparicio, A. Suárez, Existence and nonexistence of positive solutions for nonlinearelliptic singular equations with natural growth . Nonlinear Anal. (2013), 157–169.[19] J. Carmona, A. Molino, L. Moreno-Mérida, Existence of a continuum of solutions for a quasilinear ellipticsingular problem.
J. Math. Anal. Appl. (2016), no. 2, 1048–1062.[20] L. Damascelli, F. Gladiali,
Some nonexistence results for positive solutions of elliptic equations in unboundeddomains.
Rev. Mat. Iberoamericana (2004), no. 1, 67–86. [21] D.G. de Figueiredo, P.-L. Lions, R.D. Nussbaum, A priori estimates and existence of positive solutions ofsemilinear elliptic equations.
J. Math. Pures Appl. (9) (1982), no. 1, 41–63.[22] R. Durastanti, Asymptotic behavior and existence of solutions for singular elliptic equations.
Annali diMatematica Pura ed Applicata. (2019) https://doi.org/10.1007/s10231-019-00906-0.[23] A. Farina, L. Montoro, G. Riey, B. Sciunzi,
Monotonicity of solutions to quasilinear problems with a first–order term in half–spaces.
Ann. Inst. H. Poincaré Anal. Non Linéaire (2015), no. 1, 1–22.[24] R. Filippucci, P. Pucci, P. Souplet, A Liouville-type theorem for an elliptic equation with superquadraticgrowth in the gradient.
Adv. Nonlinear Stud. (2020), no. 2, 245–251.[25] B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations.
Comm. PartialDifferential Equations (1981), no. 8, 883–901.[26] B. Gidas, J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations.
Comm.Pure Appl. Math. (1981), no. 4, 525–598.[27] M.A. Krasnosel’ski˘ı, Fixed points of cone–compressing or cone–extending operators.
Soviet Math. Dokl. (1960) 1285–1288.[28] O. Ladyzhenskaya, N. Ural’tseva, Linear and quasilinear elliptic equations.
Translated from the Russian byScripta Technica,
Academic Press, New York-London (1968), xviii+495 pp.[29] J. Li, J. Yin, Y. Ke,
Existence of positive solutions for the p − Laplacian with p − gradient term. J. Math.Anal. Appl. (2011), no. 1, 147–158.[30] P.-L. Lions,
Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre.
J. AnalyseMath. (1985), 234–254.[31] S. López-Martínez, A singularity as a break point for the multiplicity of solutions to quasilinear ellipticproblems.
Adv. Nonlinear Anal. (2020), 1351–1382.[32] P.J. Martínez-Aparicio, Singular Dirichlet problems with quadratic gradient.
Boll. Unione Mat. Ital. (9) (2009), no. 3, 559–574.[33] E. Mitidieri, S.I. Pokhozhaev, Absence of positive solutions for quasilinear elliptic problems in R N . Tr. Mat.Inst. Steklova (1999), Issled. po Teor. Differ. Funkts. Mnogikh Perem. i ee Prilozh. 18, 192–222.[34] L. Montoro,
Harnack inequalities and qualitative properties for some quasilinear elliptic equations.
NonlinearDiffer. Equ. Appl. (2019), paper no. 45, 33 pp.[35] L. Orsina, J.-P. Puel, Positive solutions for a class of nonlinear elliptic problems involving quasilinear andsemilinear terms.
Comm. Partial Differential Equations (2001), no. 9–10, 1665–1689.[36] S.I. Pohozaev, On the eigenfunctions of the equation ∆ u + λf ( u ) = 0 . Dokl. Akad. Nauk SSSR (1965),36–39.[37] A. Porretta, L. Véron,
Asymptotic behaviour of the gradient of large solutions to some nonlinear ellipticequations.
Adv. Nonlinear Stud. (2006), no. 3, 351–378.[38] J. Serrin, H. Zou, Existence and nonexistence results for ground states of quasilinear elliptic equations.
Arch.Rational Mech. Anal. (1992), no. 2, 101–130.[39] P. Souplet,
Finite time blow–up for a non–linear parabolic equation with a gradient term and applications.
Math. Methods Appl. Sci. (1996), no. 16, 1317–1333.[40] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients dis-continus.
Ann. Inst. Fourier (Grenoble) (1965), no. 1, 189–258.[41] W. Zhou, X. Wei, X. Qin, Nonexistence of solutions for singular elliptic equations with a quadratic gradientterm.
Nonlinear Anal.75