Some elliptic problems with singular nonlinearity and advection for Riemannian manifolds
aa r X i v : . [ m a t h . A P ] J a n SOME ELLIPTIC PROBLEMS WITH SINGULAR NONLINEARITYAND ADVECTION FOR RIEMANNIAN MANIFOLDS
JO ˜AO MARCOS DO ´O AND RODRIGO G. CLEMENTE
Abstract.
We are interested in regularity properties of semi-stable solutions for a classof singular semilinear elliptic problems with advection term defined on a smooth boundeddomain of a complete Riemannian manifold with zero Dirichlet boundary condition.We prove uniform Lebesgue estimates and we determine the critical dimensions forthese problems with nonlinearities of the type Gelfand, MEMS and power case. As anapplication, we show that extremal solutions are classical whenever the dimension of themanifold is below the critical dimension of the associated problem. Moreover, we analyzethe branch of minimal solutions and we prove multiplicity results when the parameter isclose to critical threshold and we obtain uniqueness on it. Furthermore, for the case ofRiemannian models we study properties of radial symmetry and monotonicity for semi-stable solutions.Mathematics Subject Classifications: Primary 35J60, Secondary 35B65, 35B45Keywords: Nonlinear PDE of elliptic type, Singular nonlinearity, Advection, Semi-stableand extremal solutions. Introduction
Let ( M , g ) be a complete Riemannian manifold with dimension N , Ω ⊂ M a smoothbounded domain and A ( x ) a smooth vector field over Ω. In the present paper, weinvestigate the following class of nonlinear elliptic differential equations involving singularnonlinearities and advection − ∆ g u + A ( x ) · ∇ g u = λf ( u ) in Ω ,u > ,u = 0 on ∂ Ω , ( P λ )We analyse ( P λ ) for the following types of nonlinearities: Mathematics Subject Classification.
Key words and phrases.
Nonlinear PDE of elliptic type, Singular nonlinearity, Advection, semi-stablesolution, Extremal solution, Regularity.Research partially supported by the National Institute of Science and Technology of Mathematics INCT-Mat, CAPES and CNPq. ( i ) f ( s ) = e s (Gelfand)( ii ) f ( s ) = (1 + s ) m , m > iii ) f ( s ) = 1 / (1 − s ) (MEMS) (1.1)The main purpose of this paper is to study the minimal branch and regularity propertiesfor minimal solutions of ( P λ ). We first prove that there exists some positive finite criticalparamater λ ∗ such that for all 0 < λ < λ ∗ the problem ( P λ ) has a smooth minimal stablesolution u λ while for λ > λ ∗ there are no solutions of ( P λ ) in any sense (cf. Theorems 1.1).We determine the critical dimension N ∗ for this class of problems, precisely we prove thatthe extremal solution of ( P λ ) is regular for N ≤ N ∗ and it is singular for N > N ∗ . We seethat the critical dimension depends only on the nonlinearity f ( s ) and does not depend ofthe Riemanian manifold M (cf. Theorem 1.2 and Table 1.4). For that, we establish L ∞ estimates, which are crucial in our argument to obtain regularity of the extremal solutions.We also prove multiplicity of solutions near the extremal parameter and uniqueness on it(cf. Theorem 1.3 and Theorem 1.4). Moreover, we prove radial symmetry and monotonicityfor semi-stable solutions of ( P λ ) if Ω = B R is a geodesic ball of a Riemannian model M (cf. Theorem 1.5).1.1. Statement of main results.
Before we state our main results we recall somestandard notations and definitions related with problem ( P λ ). Next we are assuming thefollowing values for s , which depends of the type of considered nonlinearity, precisely,( i ) s = + ∞ if f ( s ) = e s (Gelfand)( ii ) s = + ∞ if f ( s ) = (1 + s ) m (Power-type)( iii ) s = 1 if f ( s ) = 1 / (1 − s ) (MEMS) Classical solution : u ∈ C (Ω) ∩ C (Ω) is a classical solution of ( P λ ) if it solves ( P λ ) in theclassical sense (i.e. using the classical notion of derivative). Weak solution : u ∈ W , (Ω) is a weak solution of ( P λ ) if 0 ≤ u < s almost everywhere inΩ and u = s in a subset with measure zero such that f ( u ) ∈ L (Ω) and Z Ω ( ∇ g u · ∇ g φ + φA · ∇ g u ) d v g = λ Z Ω f ( u ) φ d v g , ∀ φ ∈ W , (Ω) . (1.2)We also consider weak subsolution ( weak supersolution ) in analogy with this definition. Forinstance, u ∈ W , (Ω) is a weak subsolution of ( P λ ) if 0 ≤ u < s almost everywhere inΩ and u = s in a subset with measure zero such that f ( u ) ∈ L (Ω) with “ ≤ ” (“ ≥ ”)instead of “ = ” in (1.2). Minimal solution : For problem ( P λ ), we say that a weak solution u ∈ W , (Ω) is aminimal solution if u ≤ v almost everywhere for all v supersolution. We denote minimalsolution of ( P λ ) by u λ . LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 3
Regular solution : We say that a weak solution u of ( P λ ) is a regular solution ifsup Ω u < s . Semi-stable solution : We say that a classical solution u of ( P λ ) is semi-stable solutionprovided that Z Ω (cid:0) |∇ g ξ | + ξA ( x ) · ∇ g ξ (cid:1) d v g ≥ Z Ω λf ′ ( u ) ξ d v g , ∀ ξ ∈ C (Ω) . (1.3)Analogously one defines stable solution if we have the strict inequality in (1.3). We saythat a classical solution u of ( P λ ) is unstable if u is not semi-stable.We can now formulate our main results. Using some ideas in [6, 27], we prove theexistence of a critical parameter λ ∗ which is related with the solvability of ( P λ ). Moreover,we obtain upper and lower estimates for this critical parameter λ ∗ . This implies that theexplosion threshold cannot drop arbitrarily close to zero, no matter what the field φ is.Different from [4], here we do not assume incompressibility of the flow, that is, ∇ · φ = 0. Theorem 1.1.
There exists a critical parameter λ ∗ ∈ R , λ ∗ > such that ( i ) : For all λ ∈ (0 , λ ∗ ) problem ( P λ ) possesses an unique minimal classical solution u λ which is positive and semi-stable, and the map λ → u λ ( x ) is increasing on (0 , λ ∗ ) for each x ∈ Ω . ( ii ) : The following estimates hold β (1 − β max Ω w ) ≤ λ ∗ ≤ λ , where w and β are given in Lemma 3.1 and λ is the first eigenvalue of − ∆ g + A ·∇ g with zero Dirichlet boundary condition. ( iii ) : For λ > λ ∗ there are no solutions, even in weak sense. ( iv ) : semi-stable solutions of ( P λ ) are necessarily minimal solutions. In view of item ( i ) of Theorem 1.1, we can define the function u ∗ ( x ) := lim λ ր λ ∗ u λ ( x ) , which is measurable, since it is a limit of measurable functions. If u ∗ is a weak solution of( P λ ) at λ = λ ∗ it will be called extremal solution.An important question which has attracted a lot of attention is whether the extremalsolution u ∗ is a classical solution. Here we are going to prove regularity of the extremalsolution u ∗ if the dimension of M is below the critical dimension N ∗ . Here we stress thefact that the critical dimension depends only on the nonlinearity f ( s ) and does not dependof the manifold M , which is given precisely by,( i ) N ∗ = 10 if f ( s ) = e s (Gelfand)( ii ) N ∗ = 11 if f ( s ) = (1 + s ) m (Power-type)( iii ) N ∗ = 8 if f ( s ) = 1 / (1 − s ) (MEMS) (1.4) Theorem 1.2.
The extremal solution u ∗ of ( P λ ∗ ) is classical provided that ≤ N < N ∗ . LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 4
Remark 1.1.
To obtain regularity of the extremal solutions defined on domains of R N usually make use of an argument based on Hardy-type inequality (see Subsection 1.2). It iswell known that a complete open Riemannian manifold with non-negative Ricci curvature ofdimension greater than or equal to three in which a Hardy inequality are satisfied are closeto the Euclidean space. In view of this, to obtain regularity results of extremal solutionsfor any Riemannian manifold one have to use an argument free of Hardy-type inequality. We now study uniqueness at the critical parameter λ ∗ . Theorem 1.3.
For dimension ≤ N < N ∗ , the extremal solution u ∗ is the unique classicalsolution of ( P λ ∗ ) among all weak solutions. Next, we present a multiplicity result for this class of equations when λ is smaller andclose to the critical parameter λ ∗ . The proof is carried out by the Mountain Pass Theoremin the same spirit of [15]. Theorem 1.4.
Let ≤ N < N ∗ and A = ∇ g a . Then, there exists δ > such that for any λ ∈ ( λ ∗ − δ, λ ∗ ) we have a second branch of solutions U λ given by mountain pass for J ǫ,λ on W , (Ω) . Remark 1.2.
We can use elliptic estimates to see that any regular solution u of ( P λ ) belongs to C ,α (Ω) . For that, we cover M by coordinate neighbourhood and consider apartition of unity subordinate to this cover. By using Schauder estimates it is easy toprove that u ∈ C ,α (Ω) and consequently any regular solution of ( P λ ) is a classical solution(see [17, 20]). Remark 1.3.
The class of semi-stable solutions includes local minimizers, minimalsolutions, extremal solutions and certain class of solutions found between a sub and asupersolution.
We also obtain qualitative properties for semi-stable solutions of problem ( P λ ) if Ω = B R is a geodesic ball of a Riemannian model and A is a radial vector field. Precisely, we provethat such solutions are radially symmetric and decreasing. We say that u ∈ C ( B R ) isradially symmetric and decreasing if u ( x ) = u ( r ) , where r = dist( x, O ) and u r ( r ) < r ∈ (0 , R ).The class of Riemannian model ( M , g ) includes the classical space forms. Precisely, amanifold M of dimension N ≥ O and whose metric g is given, in polarcoordinates around O , byd s = d r + ψ ( r ) d θ for r ∈ (0 , R ) and θ ∈ S N − , (1.5)where r is by construction the Riemannian distance between the point P = ( r, θ ) to thepole O , ψ is a smooth positive function in (0 , R ) and d θ is the canonical metric on theunit sphere S N − . Note that our results apply to the important case of space forms, i.e.,the unique complete and simply connected Riemannian manifold of constant sectional LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 5 curvature K ψ corresponding to the choice of ψ namely,( i ) ψ ( r ) = sinh r, K ψ = − ii ) ψ ( r ) = r, K ψ = 0 (Euclidean space)( iii ) ψ ( r ) = sin r, K ψ = 1 (Elliptic space)Next, we state a radial symmetry result for semi-stable solutions defined on a geodesicball of M . The proof is based on the fact that any angular derivative of u would be either asign changing first eigenfunction of the linearized operator at u or identically zero, thanksto the semistability condition. In addiction, the monotonicity of u is due to the positivityof the nonlinearity. Theorem 1.5. If u ∈ C ( B R ) is a classical stable solution of ( P λ ) with a radial vectorfield A , then u is radially symmetric and decreasing. We can improve the result of Theorem 1.2 giving an estimate for the radial case.
Theorem 1.6.
Let u be the extremal solution of ( P λ ∗ ) on a geodesic ball B r of a Riemannianmodel with ≤ N < N ∗ . Then u ∗ is a classical solution and k u ∗ k ∞ ≤ c, where c > is a constant which does not depends of λ . We emphasize that, for the case f ( u ) = 1 / (1 − u ) we have c < . Motivation and previous results.
In order to motivate our results we begin bygiving a brief survey on this subject. Singular elliptic problems of the form − Lu = λh ( u ) , (1.6)for a second order elliptic operator L under various boundary conditions, has beenextensively studied since the papers of D. Joseph and T. Lundgren [21], J. Keener andH. Keller [22] and M. Crandall and P. Rabinowitz [12, 13]. It has been shown in thesepioneering works that there exists a critical threshold λ ∗ > < λ < λ ∗ , while no positive solutions exist for λ > λ ∗ . In [27], F. Mignot andJ-P. Puel studied regularity results to certain nonlinearities, namely, h ( s ) = e s , g ( s ) = s m with m > g ( s ) = 1 / (1 − s ) k with k >
0. Very recently, this analysis was completed byN. Ghoussoub and Y. Guo [18] for the MEMS case, precisely, − ∆ u = λf ( x ) / (1 − u ) ina bounded domain Ω under zero Dirichlet boundary condition, among other results theyproved that N = 8 is the critical dimension for this class of problems.H. Brezis and J. L. Vazquez in [6] treated the delicate issue of regularity of solutions atextreme value λ = λ ∗ of − ∆ u = λh ( u ) , u = 0 , ∂ Ω , (1.7)where the nonlinerity h ( s ) is continuous, positive, increasing and convex function definedfor u ≥ h (0) > s →∞ h ( s ) /s = + ∞ . Typical examples are h ( s ) = e s and h ( s ) = (1 + s ) p , with p >
1. The authors characterized the singular H extremal solutionsand the extremal value by a criterion consisting of two conditions: ( i ) they must be energysolutions, not in L ∞ ; ( ii ) they must satisfy a Hardy inequality which translates the fact LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 6 that the first eigenvalue of the linearized operator is nonnegative. In order to apply thischaracterization to those examples, they also establish a simultaneous generalization forHardy’s and Poincare’s inequalities for all dimensions N ≥
2. One of main theorem in [6]is focused on the application to the cases h ( s ) = e s and h ( s ) = (1 + s ) m , m >
1, in theunit ball of Euclidean space centred at the origin. If h ( s ) = e s , the procedure shows that u ∗ ( x ) = log | x | − is an unbounded extremal solution in H for λ ∗ = 2( n −
2) if and only if n ≥
10. A similar result is also obtained for the case h ( s ) = (1 + s ) m , m > h ( s ) and domains Ω, regularity for solutions of (1.7) at λ = λ ∗ has been established by G. Nedev in [28]. Precisely, he proved that u ∗ ∈ L ∞ (Ω) if N ≤
3, while u ∗ ∈ H (Ω) if N ≤
5, for every bounded domain Ω and convex nonlinearity h ( s ) satisfying h ∈ C , nondecreasing, h (0) > , and lim s → + ∞ h ( s ) s = + ∞ . (1.8)After that X. Cabr´e in [9] proved regularity if N ≤ ≤ N ≤
9. In [9, 10] were not assumedconvexity on the nonlinearity h ( s ), but in contrast with Nedev’s result, it was assumedconvexity of the set Ω.The following equation has often been used to model a simple electrostatic Micro-Electro-Mechanical system (MEMS) device: − ∆ u = λf ( x )(1 − u ) in Ω ,u = 0 on ∂ Ω , (1.9)where Ω is a smooth bounded domain in R N , f ∈ C α (Ω), λ > < u ( x ) < < λ < λ ∗ was proved as well as the existence and uniqueness ofthe extremal solution for λ = λ ∗ provided that dimension 1 ≤ N ≤
7. In this dimensionalrange, the existence of non minimal solutions was obtained in [14], where the authors studythe branch of semi-stable solutions and where existence results in higher dimensions, for asuitable hypothesis, were also established.About the explosion problem in an incompressible flow we refer to the very interestingwork of Berestycky et al. [4] which considered the non-selfadjoint elliptic problem ( − ∆ u + φ · ∇ u = λg ( u ) in Ω ,u = 0 on ∂ Ω , (1.10) LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 7 with a prescribed incompressible flow φ ( x ) so that ∇ · φ = 0 where Ω is a smooth boundeddomain of Euclidean space. In this work the authors began to investigate how the presenceof an underlying flow and its properties affect the explosion. They were interested inqualitative dependence of the critical explosion λ ∗ with respect to the vector field φ . Asobserved numerically in [5] for a two dimensional cellular flow, the explosion thresholdincreases for flows oscillating on a small scale and it may actually decrease if the flow haslarge scale variations. This analysis motivated our result about upper and lower estimatesfor λ ∗ for a general vector field A (cf. Theorem 1.1).Still on the non-selfadjoint elliptic problem (1.10), C. Cowan and N. Ghoussoub [8]proved regularity results for extremal solutions for the nonlinearities: f ( u ) = 1 / (1 − u ) or f ( u ) = e u and a general class of advection term (not necessarily incompressible). Theargument in [8] was based on a class of Hardy type inequality contained in [11]. At thispoint we emphasize that a similar argument can not be applied for a general Riemannianmanifold setting to prove regularity (cf. Theorem 1.2), since it is known that the existenceof Hardy or Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg inequality on a Riemannianmanifold implies qualitative properties on the Riemannian manifold. Precisely, it wasshown that if ( M , g ) is a complete Riemannian manifold with nonnegative Ricci curvaturein which a Hardy or Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg type inequalitiesholds then M is close to Euclidean space in some suitable sense, see [26].X. Luo, D. Ye and F. Zhou in [25] studied (1.10) where h : [0 , a ) → R + with fixed a ∈ (0 , + ∞ ) satisfies the following condition: h is C , positive, nondecreasing and convex in [0 , a ) with lim s → a − h ( s ) = + ∞ . (H)The authors in [25] observed a close similarity between (1.10) and the Emden-Fowlerequation with superlinear regular nonlinearity, ( − ∆ u = λh ( u ) in Ω ,u = 0 on ∂ Ω , (1.11)with λ > h : [0 , + ∞ ) → [0 , + ∞ ) such that h is C , nondecreasing convex and lim s → + ∞ h ( s ) s = + ∞ . (1.12)The problem (1.11) can be linked to (1.10) where g : [0 , a ) → R + with fixed a ∈ (0 , + ∞ )satisfies (H). Without loss of generality, we can fix a = 1 and consider the transformation v = − ln(1 − u ). Thus let u solving (1.10) then v verifies ( − ∆ v + |∇ v | + c ( x ) · ∇ v = λe v g (1 − e − v ) := λh ( v ) in Ω ,v = 0 on ∂ Ω , (1.13)Therefore h satisfies (1.12) and v ∗ = − ln(1 − u ∗ ) is the extremal solution for the problem(1.13). Thus the regularity of u ∗ is equivalent to the boundedness of v ∗ . We mention thatthe situation could be very different with the presence of advection terms, see [8,31]. If thevector field φ nontrivial the operator − ∆ + φ · ∇ is not self-adjoint. However if φ = −∇ γ in LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 8
Ω then − ∆ + φ · ∇ can be rewritten as a self-adjoint operator of the form − e − γ div( e γ ∇ ). Inthat case, (1.10) admits a variational structure and we can expect more precise estimatesof minimal solutions u λ , as in the radial case.Our Theorems 1.1 and 1.3 improve and complement some results in [25] for thenonlinearities described in (1.1). In Theorem 1.2 we determine the critical dimension N ∗ and prove regularity for extremal solutions if N < N ∗ . This theorem is close relatedwith Theorem 1.3, 1.4 and 1.5 in [25] where X. Luo et al investigated the regularity ofextremal solutions of (1.10) and it was proved that if f satisfies (H) and some additionalassumptions, then u ∗ is regular if N ≤ N , where N depends on f and in the mostsignificant cases is less than 10. In our Theorem 1.6 we do not required that the advectionterm φ is the gradient of a smooth radial function, even for N = 2, to prove our regularityresult for radial case for the nonlinearities described in (1.1), differently of Theorem 1.1in [25].In the past decades, there have been considerable attentions to be paid on the researchof singular elliptic problems defined on Riemannian manifolds. A. Farina, L. Mari andE. Valdinoci [16] studied Riemannian manifolds with non-negative Ricci curvature thatposses a stable, nontrivial solution of a semilinear equation − ∆ g u = f ( u ) . (1.14)Under suitable assumptions, it was proved symmetry results for the solutions and therigidity of the underlining manifold. E. Berchio, A. Ferrero and G. Grillo in [3] studiedexistence, uniqueness and stability of radial solutions of (1.14) for the Lane-Emden-Fowlerequation, that is, f ( s ) = | s | p − s , on a Riemannian manifold of dimension N ≥ B R with zero Dirichlet boundary conditionof a Riemannian model M . They proved radial symmetry and monotonicity for the classof semi-stable solutions. Moreover, they establish L ∞ , L p and W ,p estimates which areoptimal and do not depend on the nonlinearity f ( s ). As an application, under standardassumptions on the nonlinearity f ( s ), they proved that the extremal solution u ∗ is boundedwhenever N ≤ Outline.
The paper is organized as follows. In the next section we bring a version ofMaximum principle, we prove a Sub- and Super-solution method and a version of Hodge-Helmholtz decomposition. In Section 3, we study the existence of extremal parameter λ ∗ and minimal solutions u λ . The Section 4 is devoted to prove monotonicity results for thebranch of minimal solutions and L p -estimates for u λ uniformly in λ , and we determinethe critical dimensions for this class of problems for singular nonlinearities of type MEMS,Gelfand and power case. By using this estimates, we prove that the extremal solution u ∗ := lim λ ր λ ∗ u λ is classical whenever the dimension of M is below the critical dimension. LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 9
In Section 5, for the particular case where M is a Riemannian model and Ω is a geodesicball of M , we establish symmetry and monotonicity for the class of semi-stable solutionsand we also prove L ∞ -estimates for u ∗ . In Section 6 we analyze the branch of minimalsolutions and we prove multiplicity of solutions if λ ∈ ( λ ∗ − δ, λ ∗ ) for some δ > λ ∗ . Key-ingredients
We use a Comparison Principle for weak solutions of quasilinear elliptic differentialequation in divergence form on complete Riemannian manifold. We need a simple versionof Theorem 3.3 found in [2].
Proposition 2.1 (Maximum Principle) . Let w a weak supersolution of − ∆ g u + A · ∇ u = 0 .If w ≥ on ∂ Ω , then w ≥ in Ω . For the sake of completeness, we prove the Sub and Supersolution result in Proposition2.2 using the Monotone Iteration Method. In this way, T. Kura [24] has proved manyresults about the existence of a solution between sub and supersolutions for quasilinearproblems.
Proposition 2.2 (The sub- and super-solution method) . Let u and u subsolution andsupersolution of ( P λ ) , respectively, that satisfies u ≤ u a.e. in Ω . Then problem ( P λ ) hasa weak solution u such that u ≤ u ≤ u a.e. in Ω .Proof. Denote by u = u . We define a sequence u n inductively where each u n is the uniqueweak solution of the problem ( − ∆ g u n + A · ∇ g u n + cu n = λf ( u n − ) + cu n − in Ω ,u n = 0 on ∂ Ω . (2.1)This sequence satisfies u ≤ u n − ≤ u n ≤ u . In fact, consider (2.1) where n = 1 . Wehave u ∈ W , (Ω) and by Maximum Principle follows u ≤ u ≤ u. In the same way u ∈ W , (Ω) and satisfies u ≤ u ≤ u ≤ u . By induction we have the result i.e., u ≤ u n − ≤ u n ≤ u . Now, observe that u n is bounded in W , (Ω) and has a subsequencethat converges weakly to u ∈ W , (Ω). Taking the limit in the equation follows that u isa weak solution of the problem ( − ∆ g u + A · ∇ g u = λf ( u ) in Ω ,u = 0 on ∂ Ω . (cid:3) We also have a version of Hodge-Helmholtz decomposition in order to deal with generalvector fields A. This decomposition of vector fields is one of the fundamental theorem influid dynamics. It describes a vector field in terms of its divergence-free and rotation-freecomponents. For more results in this subject we refer the reader to [30].
LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 10
Lemma 2.1.
Any vector field A ∈ C ∞ (Ω , T M ) can be decomposed as A = −∇ g a + C where a is a smooth scalar function and C is a smooth bounded vector field such that div( e a C ) = 0 .Proof. Let ν the unit outer normal on ∂ Ω . Using Krein-Rutman theorem, we can find apositive solution w of ( ∆ g w + div( wA ) = µw in Ω , ( ∇ g w + wA ) · ν = 0 on ∂ Ω . for a constant µ ∈ R . Integrating the equation over Ω one sees that µ = 0 . By the maximumprinciple, w is positive up to the boundary. Now define a := log( w ) and C := A + ∇ g a .It is easy to see that div( e a C ) = e a ∇ g a · A + e a div C = ∇ g w · A + |∇ g w | /w + w div A −|∇ g w | /w + ∆ w = 0 . (cid:3) Existence results
Now we can construct a supersolution for the problem ( P λ ) if λ is sufficient small. Lemma 3.1.
Let w ∈ W , (Ω) be a weak solution of the problem (cid:26) − ∆ g w + A · ∇ g w = 1 in Ω ,w = 0 on ∂ Ω . (3.1) There exist β > such that βw is a supersolution of ( P λ ) for λ sufficient small.Proof. For a large c >
0, let ˜ L w = − ∆ g w + A · ∇ g w + cw and consider the problem ( ˜ L w = f in Ω ,w = 0 on ∂ Ω . If we write w = ˜ L − (1 + cw ) = N ( w ) we can use Schauder Fixed Point Theorem to finda solution w of (3.1). By elliptic estimates w ∈ C (Ω) so we can take β > β max Ω w < s . If λ ≤ β/f ( β max Ω w ) we have Z Ω ( ∇ g ( βw ) ∇ g φ + φA · ∇ g ( βw )) d v g = β Z Ω φ ≥ Z Ω λφf ( βw ) d v g , i.e., βw is a supersolution of ( P λ ). (cid:3) Let us define Λ := { λ ≥ P λ ) has a classical solution } . We can define the extremal parameter λ ∗ = sup Λ . Remark 3.1.
Using Lemma 3.1 we can find a regular solution between and βw . Withthis, sup Λ > . Lemma 3.2.
The set Λ is a interval. LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 11
Proof.
Initially, we prove that Λ does not consist of just λ = 0. Let u a classical solutionfor problem ( P λ ) with λ < λ ∗ . Observe that u = 0 and u are sub and supersolution,respectively, for the problem ( P λ ). Using the Sub and Supersolution Method, there exista weak solution v ∈ W , (Ω) such that u ≤ v < u < s . By Remark 1.2, v is a classicalsolution. This solution is a supersolution for ( P µ ) if µ ∈ (0 , λ ). Again, there exist aclassical solution for the problem ( P µ ). Thus, Λ is a interval. (cid:3) Lemma 3.3.
The interval Λ is bounded.Proof. Suppose that exist a classical solution u of ( P λ ), for λ sufficiently large. Wecan suppose that λ > λ , where λ is the first eigenvalue associate to the operator L = − ∆ g + A · ∇ g . Let v the first eigenvalue in λ , i.e., (cid:26) − ∆ g v + A · ∇ g v = λ v in Ω ,v = 0 on ∂ Ω . By regularity theory, follows that v ∈ C ,α (Ω). By homogeneity, we can suppose k v k ∞ < . So v and u satisfies − ∆ g v + A · ∇ g v = λ v < λf ( u ) = − ∆ g u + A · ∇ g u. By Comparison Principle follows that v ≤ u. Now, given ǫ >
0, we take v a solution of ( − ∆ g v + A · ∇ g v = ( λ + ǫ ) v in Ω ,v = 0 on ∂ Ω . As above, v ≤ v ≤ u . By induction, we have solutions v n such that ( − ∆ g v n + A · ∇ g v n = ( λ + ǫ ) v n − in Ω ,v n = 0 on ∂ Ω , with v ≤ ... ≤ v n − ≤ v n ≤ u in C ,α (Ω). So, v n ⇀ v in W , (Ω). It follows that v satisfies ( − ∆ g v + A · ∇ g v = ( λ + ǫ ) v in Ω ,v = 0 on ∂ Ω . This is impossible since the first eigenvalue is isolated. (cid:3)
Remark 3.2.
Clearly, λ ∗ < + ∞ and there are no classical solution of ( P λ ) for λ > λ ∗ . Monotonicity results and estimates for minimal solutions
Minimal solutions.Lemma 4.1.
For each λ < λ ∗ there exist a unique minimal solution u λ for the problem ( P λ ) . Therefore, for all x ∈ Ω , the map λ → u λ ( x ) is strictly increasing.Proof. Consider the weak solution u given by Proposition 2.2. All supersolutions v of ( P λ )satisfies u ≤ v. Thus u is minimal. The uniqueness follows by minimality of u . In this way,we define u := u λ . Therefore, if λ < µ, we have that u µ is a supersolution of ( P λ ). Thus, u λ < u µ . (cid:3) LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 12
Let u be a semi-stable solution of ( P λ ), and let us consider the following eigenvalueproblem involving the linearized operator L u,λ = − ∆ g + A · ∇ g − λf ′ ( u ) at u , (cid:26) L u,λ φ = µφ in Ω u = 0 on ∂ Ω . It is well known that there exists a smallest positive eigenvalue µ , which we denote by µ ,λ ,and an associated eigenfunction φ ,λ > µ ,λ is a simple eigenvalue and has thefollowing variational characterization µ ,λ = inf (cid:26) h L u,λ φ, φ i L (Ω) : φ ∈ W , (Ω) , Z Ω φ d v g = 1 (cid:27) . Lemma 4.2. If ≤ λ < λ ∗ , the minimal solutions are semi-stable.Proof. Let u λ minimal solution of ( P λ ). Suppose that u λ is not semi-stable i.e., the firsteigenvalue µ ,λ of operator L u,λ is negative. Consider the function ψ ǫ = u λ − ǫψ ∈ W , (Ω),where ψ ∈ W , (Ω) is the first positive eigenvalue of operator − ∆ g + A · ∇ g − λf ′ ( u λ ).Using Taylor’s formula, for k ξ k W , (Ω) sufficiently small we have − ∆ g ψ ǫ + A · ∇ g ψ ǫ − λf ( ψ ǫ ) = − ǫλf ( ψ ) + λf ( u λ ) − ǫκψ − ǫλf ′ ( u λ ) ψ = − ǫκψ − λǫ f ′′ ( ξ ) ψ ≥ , for ǫ sufficiently small, because κ < . Thus ψ ǫ is a supersolution of ( P λ ) and, by minimalityof u λ we have a contradiction. (cid:3) Proof of Theorem 1.1. (1) The existence of λ ∗ follows from Lemma 3.3. By Lemmas 4.1and 4.2, there exists u λ minimal solution of ( P λ ) which is semi-stable and the function λ → u λ ( x ) is strictly increasing.(2) Note that since u = 0 is a subsolution of ( P λ ), u λ is non negative. In the same way,since a classical solution of ( P λ ) is also a supersolution, it follows that u λ is a classicalsolution. The estimate is a consequence of Lemma 3.1 and Lemma 3.3.(3) Let u µ a weak solution of ( P µ ) with λ ∗ < µ . Observe that w = (1 − ǫ ) u µ is a weaksolution of − ∆ g w + A · ∇ g w = (1 − ǫ ) µf ( u µ ), that is, Z Ω ( ∇ g w · ∇ g φ + φA · ∇ g w ) d v g = (1 − ǫ ) µ Z Ω φf ( u µ ) d v g . An easy calculation shows that w is a supersolution for ( P (1 − ǫ ) µ ). Thus there exist a weaksolution v ≤ w . Since v ≤ w < u µ , it follows that v is a classical solution of ( P (1 − ǫ ) µ ). If ǫ is sufficiently small, λ ∗ < (1 − ǫ ) µ . But this is a contradiction. Furthermore, since u ∗ is amonotone limit of measurable functions, it is also measurable.(4) Now, to prove that a semi-stable solution of ( P λ ) is minimal, let u and v a semi-stablesolution and a supersolution of ( P λ ) respectively. For θ ∈ [0 ,
1] and 0 ≤ φ ∈ W , (Ω), we LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 13 have I θ,φ := Z Ω ( ∇ g ( θu + (1 − θ ) v ) · ∇ g φ + φA · ∇ g ( θu + (1 − θ ) v )) d v g − λ Z Ω φf ( θu + (1 − θ ) v ) d v g ≥ , due to the convexity of function s → f ( s ) . Since I ,φ = 0 , the derivative of I θ,φ at θ = 1 isnon positive, that is Z Ω ( ∇ g ( u − v ) · ∇ g φ + φA · ∇ g ( u − v )) d v g − Z Ω λ ( u − v ) φf ′ ( u ) d v g ≤ , ∀ φ ≥ . Testing φ = ( u − v ) + and using that u is semi-stable we get that Z Ω (cid:0) |∇ g ( u − v ) + | + ( u − v ) + A · ∇ g ( u − v ) (cid:1) d v g − Z Ω λ ( u − v )( u − v ) + f ′ ( u ) d v g = 0 , for all φ ≥
0. Since I θ, ( u − v ) + ≥ θ ∈ [0 ,
1] and I , ( u − v ) + = ∂ θ I , ( u − v ) + = 0 , we have ∂ θθ I , ( u − v ) + = − Z Ω λ ( u − v ) ( u − v ) + f ′′ ( u ) d v g ≥ . Clearly we have ( u − v ) + = 0 a.e. in Ω and therefore R Ω |∇ g ( u − v ) + | d v g = 0, from whichwe conclude that u ≤ v a.e. in Ω. (cid:3) Determining the critical dimension.
For the MEMS case, the next lemma is theprincipal estimate, which was already behind the proof of the regularity of semi-stablesolutions in dimensions lower than 7.When A = −∇ g a + C the problem ( P λ ) can be rewritten as − div g ( e a ∇ g u ) + e a C · ∇ g u = λe a (1 − u ) . Thus the semi-stability and weak solution conditions becomes, respectively Z Ω (cid:0) e a |∇ g η | + e a ηC · ∇ g η (cid:1) d v g ≥ Z Ω λe a (1 − u ) η d v g (4.1)and Z Ω ( e a ∇ g u · ∇ g φ + e a φC · ∇ g u ) d v g = Z Ω λe a φ (1 − u ) d v g . (4.2) Lemma 4.3. If u is a semi-stable solution of ( P λ ) with < λ < λ ∗ , f ( u ) = 1 / (1 − u ) and < t < √ , holds the following estimate k e a/ (2 t +3) (1 − u ) − k L t +3 / ≤ (cid:20) t + 1)2 + 4 t − t (cid:21) /t C k Ω k / (2 t +3) . LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 14
Proof.
Let 0 < t < √ u semi-stable solution of ( P λ ). Taking η := (1 − u ) − t − R Ω e a ηC · ∇ g η d v g = 0.Thus, testing η in the semistability condition (4.1), we obtain0 ≤ Z Ω e a { t (1 − u ) − t − |∇ g u | − λ (1 − u ) − (cid:2) (1 − u ) − t − (cid:3) } d v g = Z Ω e a { t (1 − u ) − t − |∇ g u | − λ (1 − u ) − t − + 4 λ (1 − u ) − t − − λ (1 − u ) − } d v g , which implies − Z Ω e a (1 − u ) − t − |∇ g u | d v g ≤ − λt Z Ω e a (1 − u ) − t − d v g + 4 λt Z Ω e a (1 − u ) − t − d v g . (4.3)Testing φ := (1 − u ) − t − − Z Ω e a |∇ g u | (2 t + 1)(1 − u ) − t − d v g = Z Ω (cid:8) λe a (1 − u ) − t − − λe a (1 − u ) − (cid:9) d v g ≤ Z Ω λe a (1 − u ) − t − d v g , (4.4)because with this choice of φ we can check that R Ω e a φ C · ∇ g u d v g = 0. Using (4.4) and(4.3) we have − t + 1 Z Ω e a (1 − u ) − t − d v g ≤ − t Z Ω e a (1 − u ) − t − d v g + 4 t Z Ω e a (1 − u ) − t − d v g and follows that (cid:18) t − t + 1 (cid:19) Z Ω e a (1 − u ) − t − d v g ≤ t Z Ω e a (1 − u ) − t − d v g . Using H¨older inequality with conjugate exponents (2 t + 3) / ( t + 3) and (2 t + 3) /t , we have (cid:18) t − t + 1 (cid:19) Z Ω e a (1 − u ) − t − d v g ≤ t k Ω k t/ (2 t +3) (cid:20)Z Ω e a (2 t +3) / ( t +3) (1 − u ) − t − d v g (cid:21) ( t +3) / (2 t +3) ≤ C t k Ω k t/ (2 t +3) (cid:20)Z Ω e a (1 − u ) − t − d v g (cid:21) ( t +3) / (2 t +3) , where C = (cid:20) sup Ω e at/ ( t +3) (cid:21) ( t +3) / (2 t +3) . Thus, (cid:18) t − t + 1 (cid:19) (cid:20)Z Ω e a (1 − u ) − t − d v g (cid:21) t/ (2 t +3) ≤ C t k Ω k t/ (2 t +3) and therefore k e a/ (2 t +3) (1 − u ) − k L t +3 / ≤ (cid:20) t + 1)2 + 4 t − t (cid:21) /t C k Ω k / (2 t +3) , LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 15 this is the desired estimate. (cid:3)
Remark 4.1.
In the above estimate we used that t + 1) > t which is an immediatelyconsequence of our assumption < t < √ . Remark 4.2.
By the above estimate, e a (1 − u λ ) − is bounded uniformly in λ over L p (Ω) for all p < p := 7 / √ . By elliptic estimates, u λ is uniformly bounded in W ,p (Ω) .Thus u ∗ is a weak solution of ( P λ ∗ ) . If we take the limit λ ր λ ∗ , we obtain the same L p estimate to extremal solution u ∗ . Proposition 4.1. If ≤ N ≤ then u ∗ is a classical solution of ( P λ ∗ ) .Proof. Note that e a (1 − u ∗ ) − ∈ L N/ (Ω). By elliptic regularity we have u ∗ ∈ W , N/ (Ω)and by Sobolev immersion u ∗ ∈ C , / (Ω) . If we suppose that k u ∗ k ∞ = 1, there exist aelement x ∈ Ω such that u ∗ ( x ) = 1. Since | − u ∗ ( x ) | ≤ C dist( x, x ) / we have, e a/ − u ∗ ( x ) ≥ Ce a/ dist( x, x ) / , and hence ∞ > Z Ω e Na/ ((1 − u ∗ ) ) N/ d v g ≥ C inf x ∈ Ω { e a/ } Z Ω x, x ) N d v g = ∞ . This is a contradiction. Thus k u ∗ k ∞ <
1. This implies e a (1 − u ∗ ) − ∈ L ∞ (Ω) and u ∗ is aclassical solution of ( P λ ∗ ). (cid:3) With a slight variation of the above arguments, the same approach works on the Gelfandand Power-type cases.
Lemma 4.4. If u is a semi-stable solution of ( P λ ) with < λ < λ ∗ , f ( u ) = e u and < t < , holds the following estimate k e / (2 t +1) a + u k L t +1 ≤ (cid:20) t t − t (cid:21) /t C k Ω k / (2 t +1) . Proof.
Let 0 < t < u semi-stable solution of ( P λ ). Define η := e tu − φ := e tu − η in the semistability condition we have0 ≤ Z Ω t e a +2 tu |∇ g u | − λe a +(2 t +1) u + 2 λe a +( t +1) u d v g , because with this choice of η we have R Ω e a ηC ∇ g η d v g = 0 . It follows that − Z Ω e a +2 tu |∇ g u | d v g ≤ − λt Z Ω e a +(2 t +1) u d v g + 2 λt Z Ω e a +( t +1) u d v g . (4.5)Testing φ in the weak solution condition we obtain2 t Z Ω e a +2 tu |∇ g u | d v g ≤ λ Z Ω e a +(2 t +1) u d v g , (4.6) LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 16 because with this choice of φ we can check that R Ω e a φC · ∇ g u d v g = 0 , because Ω is simplyconnected. Using (4.5) and (4.6) we have (cid:18) t − t (cid:19) Z Ω e a +(2 t +1) u d v g ≤ t Z Ω e a +( t +1) u d v g . Using H¨older inequality with conjugate exponents (2 t + 1) / ( t + 1) and (2 t + 1) /t , we have (cid:18) t − t (cid:19) Z Ω e a +(2 t +1) u d v g ≤ C t (cid:20)Z Ω e a +(2 t +1) u (cid:21) ( t +1) / (2 t +1) k Ω k t/ (2 t +1) d v g , where C = (cid:20) sup Ω e t/ ( t +1) a (cid:21) ( t +1) / (2 t +1) . Therefore k e / (2 t +1) a + u k L t +1 ≤ (cid:20) t t − t (cid:21) /t C k Ω k / (2 t +1) . (cid:3) Remark 4.3.
In the above estimate we used that /t − / (2 t ) > which is an immediatelyconsequence of our assumption < t < . Remark 4.4.
The above estimate said that e a + u is bounded uniformly in λ over L p (Ω) forall p < p := 4 + 1 = 5 . By elliptic estimates, u λ is uniformly bounded in W ,p (Ω) . Thus u ∗ is a weak solution of ( P λ ∗ ) . Taking the limit in λ , we obtain the same L p estimate aboveto extremal solution u ∗ . Proposition 4.2. If ≤ N ≤ then u ∗ is a classical solution of ( P λ ∗ ) .Proof. Note that e a + u ∈ L p (Ω) with p <
5. By elliptic regularity we have u ∗ ∈ W ,p (Ω) andby Sobolev immersion u ∗ ∈ C ,α (Ω) if N <
10. Thus u ∗ is a classical solution of ( P λ ∗ ). (cid:3) Lemma 4.5. If u is a semi-stable solution of ( P λ ) with < λ < λ ∗ , f ( u ) = (1 + u ) m , b > and mb − √ m ( m − b < t < mb + √ m ( m − b , holds the following estimate k e am/ [2 bt + m − (1 + u ) m k L [2 bt + m − /m ≤ (cid:18) − b t m [2 bt − (cid:19) − / [ tb ] C k Ω k / [(2 t +1) b ] . Proof.
Define η := (1 + u ) bt − φ := (1 + u ) bt − − b >
0. Testing η in thesemistability condition we have Z Ω b t e a (1 + u ) bt − |∇ g u | d v g ≥ λm Z Ω e a (1 + u ) ( m − (cid:2) (1 + u ) bt − u ) bt (cid:3) d v g , because with this choice of η we have R Ω e a ηC ∇ g η d v g = 0 . Testing φ in the weak solutioncondition we obtain(2 bt − Z Ω e a (1 + u ) bt − |∇ g u | d v g ≤ λ Z Ω e a (1 + u ) bt + m − d v g , LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 17 because with this choice of φ we can check that R Ω e a φC ∇ g u d v g = 0. It follows that (cid:18) − b t m [2 bt − (cid:19) Z Ω e a (1 + u ) bt + m − d v g ≤ Z Ω e a (1 + u ) bt + m − d v g . Using H¨older inequality with conjugate exponents bt + m − bt + m − and bt + m − bt , we have (cid:18) − b t m [2 bt − (cid:19) Z Ω e a (1+ u ) bt + m − d v g ≤ C (cid:20)Z Ω e a (1 + u ) bt + m − (cid:21) bt + m − bt + m − k Ω k bt bt + m − d v g , where C = 2 (cid:20) sup Ω e at/ ( t +1) (cid:21) t/ (2 t +1) . Therefore k e am/ [2 bt + m − (1 + u ) m k L [2 bt + m − /m ≤ (cid:18) − b t m [2 bt − (cid:19) − / [ tb ] C k Ω k / [(2 t +1) b ] . (cid:3) Remark 4.5.
The above estimate said that e a (1 + u ) m is bounded uniformly in λ over L p (Ω) for all p < − m + m p m ( m − . By elliptic estimates, u λ is uniformly bounded in W ,p (Ω) . Thus u ∗ is a weak solution of ( P λ ∗ ) . Taking the limit in λ , we obtain the same L p estimate above to extremal solution u ∗ . Proposition 4.3. If ≤ N ≤ then u ∗ is a classical solution of ( P λ ∗ ) .Proof. Since e a (1 + u ∗ ) m ∈ L p (Ω) with p < − m + m p m ( m − u ∗ ∈ W ,p (Ω) and by Sobolev immersion u ∗ is a classical solution if N < m − (cid:16)p m ( m −
1) + 1 (cid:17) . Observe that √ m ( m − m − > m > N ≤ u ∗ is a classical solution of ( P λ ∗ ). (cid:3) Proof of Theorem 1.2.
It follows from Propositions 4.1, 4.2, 4.3 (cid:3) Symmetry and monotonicity
Proof of Theorem 1.5.
Let u ∈ C ( B R ) a stable solution of ( P λ ). The stability condition(1.2) is equivalent to the positivity of the first eigenvalue of L u,λ in B R , i.e., µ ,λ = inf ξ ∈ W , ( B R ) \{ } R B R {|∇ g ξ | + ξA · ∇ g ξ − λf ′ ( u ) ξ } d v g R B R ξ d v g > . Now, consider u θ = ∂u∂θ any angular derivative of u . By the fact u ∈ C ( B R ), we have Z B R |∇ g u θ | d v g < ∞ . Moreover, the regularity up the boundary of u and the fact that u = 0 on ∂ B R give that u θ = 0 on ∂ B R . Hence, u θ ∈ W , ( B R ). Differentiate the problem ( P λ ) we obtain that u θ LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 18 weakly satisfies ( − ∆ g u θ + A · ∇ g u θ = λf ′ ( u ) u θ in B R ,u θ = 0 on ∂ B R . Multiplying the above equation by u θ and integrating by parts we have Z B R {|∇ g u θ | + u θ A · ∇ g u θ − λf ′ ( u ) u θ } d v g = 0 . It follows that either | u θ | is the first eigenvalue of linearized operator at u or u θ = 0. Butthe first alternative can not occur because µ ,λ >
0. It follows that u θ = 0 for all θ ∈ S N − .Thus u is radial. On the other hand, in spherical coordinates given by (1.5), the Laplacianoperator of u = u ( r, θ , ..., θ N − ) is given by∆ g u = 1 ψ N − ( ψ N − u r ) r + 1 ψ ∆ S N − u, where ∆ S N − is the Laplacian on the unit sphere S N − . To prove the monotonicity, notethat since u = u ( r ) and A = A ( r ), the equation ( P λ ) becomes Z s Z π e a ( ψ N − u r ) r d r d θ = Z s Z π − e a ψ N − f ( u ) d r d θ. Therefore, u r < (cid:3) Regularity in radial case.
In view of the previous section, we can write the problem( P λ ) for radial solutions u ∈ C ( B R ) as − ( e a ψ N − u r ) r + e a ψ N − C ( r ) u r = λe a ψ N − f ( u ) em (0 , R ) ,u > , R ) ,u r (0) = u ( R ) = 0 . ( R λ )In the same way, the semistability and weak solution condition becomes, respectively Z R e a ψ N − ξ r + e a ψ N − C ( r ) ξξ r d r ≥ Z R λe a ψ N − f ′ ( u ) ξ d r and Z R e a ψ N − u r φ r + e a ψ N − C ( r ) u r φ d r = Z R λe a ψ N − f ( u ) φ d r. In radial case, we obtain a more precise information about the L ∞ norm of the extremalsolution. Again, we will start with MEMS case. Lemma 5.1. If u is a classical semi-stable solution of ( R λ ) with f ( u ) = 1 / (1 − u ) , thenfor all < t < √ we have k e a/ (2 t +3) ψ N − / (2 t +3) (1 − u ) − k L t +3 / ≤ (cid:20) t + 1)2 + 4 t − t (cid:21) /t C R / (2 t +3) . LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 19
Proof.
We follow the proof of Lemma 4.3. Let 0 < t < √ u semi-stable classicalsolution of ( R λ ). Define η := (1 − u ) − t − φ := (1 − u ) − t − −
1. Applying η in thesemistablity condition we have − Z R e a ψ N − (1 − u ) − t − u r d r ≤ t Z R e a ψ N − (1 − u ) − t − d r + 4 t Z R e a ψ N − (1 − u ) − t − d r. (5.1)Applying φ in the weak solution condition, it follows that Z R e a ψ N − u r (2 t + 1)(1 − u ) − t − d r ≤ Z R e a ψ N − (1 − u ) − t − d r. (5.2)Using (5.1) and (5.2) we obtain (cid:18) t − t + 1 (cid:19) Z R e a ψ N − (1 − u ) − t − d r ≤ t Z R e a ψ N − (1 − u ) − t − d r. Using H¨older inequality with conjugate exponents (2 t + 3) / ( t + 3) and (2 t + 3) /t , (cid:18) t − t + 1 (cid:19) Z R e a ψ N − (1 − u ) − t − d r ≤ t R t/ (2 t +3) C ( t, ψ ) (cid:20)Z R e a ψ N − (1 − u ) − t − d r (cid:21) ( t +3) / (2 t +3) , where C := " sup [0 ,R ] e at/ ( t +3) ψ ( N − t/ ( t +3) ( t +3) / (2 t +3) . Thus, (cid:18) t − t + 1 (cid:19) (cid:20)Z R e a ψ N − (1 − u ) − t − (cid:21) t/ (2 t +3) ≤ t C R t/ (2 t +3) and therefore k e a/ (2 t +3) ψ N − / (2 t +3) (1 − u ) − k L t +3 / ≤ (cid:20) t + 1)2 + 4 t − t (cid:21) /t C R / (2 t +3) . (cid:3) Lemma 5.2.
Let u be a radially decreasing and semi-stable classical solution of ( R λ ) with f ( u ) = 1 / (1 − u ) . If ≤ p < ∞ , we have the estimate u (0) ≥ u ( r ) ≥ u (0) − rC k e a/p ψ ( N − /p (1 − u ) − k p . Proof.
By the Mean value theorem, there exists c ∈ (0 , r ) such that − u ( r ) + u (0) = − u ′ ( c ) r (5.3) LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 20
Integrating the equation ( R λ ) from 0 to c we obtain − e a ( c ) ψ N − ( c ) u ′ ( c ) = Z c e a ψ N − (1 − u ) − ≤ (cid:20)Z R e a ψ ( N − (1 − u ) − p (cid:21) /p (cid:20)Z R e a (1 − /p ) ψ ( N − − /p ) d r (cid:21) p/ ( p − . Using (5.3) we conclude the proof because − u ( r ) + u (0) ≤ rC k e a/p ψ ( N − /p (1 − u ) − k L p where C = e − a ( c ) ψ − N ( c ) hR R e a (1 − /p ) ψ ( N − − /p ) d r i p/ ( p − . (cid:3) Lemma 5.3.
Let u a radially decreasing and semi-stable classical solution of ( R λ ) with ≤ N ≤ . Then, for all < t < √ , we have Z r e a ψ N − D ( r ) t +3 d r ≤ (cid:18) t + 1)4 t + 2 − t (cid:19) (2 t +3) /t , where D ( r ) := 1 − k u k ∞ + C (4(2 t + 1) / (2 + 4 t − t )) /t R /p r. Proof.
Take p = t + 3 /
2. By Lemma 5.2,1 − u ( r ) ≤ − u (0) + rC k e a/p ψ ( N − /p (1 − u ) − k L p . Multiplying some positive terms and using Lemma 5.1, it follows that e − a ψ − ( N − (1 − u ( r )) t +3 ≤ e − a ψ − ( N − − u (0) + C C (cid:18) t + 1)2 + 4 t − t (cid:19) /t R /p r ! t +3 . We have Z r e a ψ N − drD ( r ) t +3 ≤ Z r e a ψ N − dr (1 − u ( r )) t +3 . where C := C C . Thus, Z r e a ψ N − drD ( r ) t +3 ≤ (cid:18) t + 1)2 + 4 t − t (cid:19) (2 t +3) /t R /p . (cid:3) We split the proof of Theorem 1.6 in three cases, namely, MEMS, Gelfand and Powercases.
Proof of Theorem 1.6 (MEMS case).
Using the Lemma 5.3, we have Z r e a ψ N − D ( r ) t +3 d r ≤ (cid:18) t + 1)4 t + 2 − t (cid:19) (2 t +3) /t R /p . (5.4) LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 21
Calculating the left-hand side above, we have Z r e a ψ N − D ( r ) t +3 d r = R r e a ψ N − d rD ( r ) t +3 + (2 t + 4) Z r D ′ R r e a ψ N − d rD t +42 d r ≥ R r e a ψ N − d rD ( r ) t +3 . (5.5)Applying (5.5) in (5.4), it follows that R r e a ψ N − d rD ( r ) t +3 ≤ (cid:18) t + 1)4 t + 2 − t (cid:19) (2 t +3) /t R /p . (5.6)Calculating the equation (5.6) and taking λ ր λ ∗ we have k u ∗ k ∞ ≤ − C, where C := (cid:20)Z R e a ψ N − d r (cid:21) / (2 t +3) (cid:0) t + 1) / (4 t + 2 − t ) (cid:1) − /t R / ( p ) − C (cid:0) t + 1) / (2 + 4 t − t ) (cid:1) /t R /p . (cid:3) With a slight variation of the above arguments, the same approach works for the Gelfandproblem with advection.
Lemma 5.4.
Let u a radially decreasing and semi-stable classical solution of ( R λ ) with f ( u ) = e u . If ≤ p < ∞ , we have the estimate u (0) ≥ u ( r ) ≥ u (0) − rC k e a/p + u ψ ( N − /p k p . Proof.
There exists c ∈ (0 , r ) such that − u ( r ) + u (0) = − u ′ ( c ) r. (5.7)Integrating the equation ( R λ ) from 0 to c we obtain − e a ( c ) ψ N − ( c ) u ′ ( c ) = Z c e a + u ψ N − ≤ (cid:20)Z R e a + up ψ N − (cid:21) /p (cid:20)Z R e a (1 − /p ) ψ ( N − − /p ) d r (cid:21) p/ ( p − . Using (5.7) we obtain − u ( r ) + u (0) ≤ C k e a/p + u ψ ( N − /p k L p r where C = e − a ( c ) ψ − N ( c ) hR R e a (1 − /p ) ψ ( N − − /p ) d r i p/ ( p − . (cid:3) LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 22
Proof of Theorem 1.6 (Gelfand case).
Take p = 2 t + 1. By Lemma 5.4 and using Lemma4.4, it follows that − u ( r ) ≤ − u (0) + C C (cid:0) (2 t ) / (2 t − t ) (cid:1) /t R /p r. Multiplying some positive terms we have e a ψ ( N − e − u (2 t +1) ≤ e a ψ ( N − e ( − u (0)+ C C ( t ) / (2 t − t ) ) /t R /p r )(2 t +1) . Thus, Z r e a ψ ( N − e ( u (0) − C ( t ) / (2 t − t ) ) /t R /p r )(2 t +1) ≤ C (cid:18) t t − t (cid:19) (2 t +1) /t R /p , where C := C C . Calculating the left-hand side above, we have e ( u (0) − C ( t ) / (2 t − t ) ) /t R /p r )(2 t +1) Z r e a ψ ( N − d r ≤ C (cid:18) t t − t (cid:19) (2 t +1) /t R /p . Taking the limit λ ր λ ∗ we have k u ∗ k ∞ ≤ ln (cid:16) C R /p (cid:0) t t − t (cid:1) (2 t +1) /t (cid:17) (2 t + 1) + C (cid:0) t ) / (2 t − t ) (cid:1) /t R /p . (cid:3) Lemma 5.5.
Let u a radially decreasing and semi-stable classical solution of ( R λ ) with f ( u ) = (1 + u ) m . If ≤ p < ∞ , we have the estimate u (0) ≥ u ( r ) ≥ u (0) − rC k e a/p ψ ( N − /p (1 + u ) m k L p . Proof.
There exists c ∈ (0 , r ) such that − u ( r ) + u (0) = − u ′ ( c ) r. (5.8)Integrating the equation ( R λ ) from 0 to c we obtain − e a ( c ) ψ N − ( c ) u ′ ( c ) = Z r e a ψ N − (1 + u ) m ≤ (cid:20)Z r e a ψ ( N − (1 + u ) mp d r (cid:21) /p (cid:20)Z r e a (1 − /p ) p ′ ψ ( N − − /p ) p ′ d r (cid:21) /p ′ . Using (5.8) we obtain − u ( r ) + u (0) ≤ C k e a/p + u ψ ( N − /p k L p r where C = e − a ( c ) ψ − N ( c ) hR R e a (1 − /p ) ψ ( N − − /p ) d r i p/ ( p − . (cid:3) LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 23
Integrating the equation ( R λ ) from 0 to r we obtain − e a ψ N − u ′ ( r ) = Z r e a ψ N − (1 + u ) m ≤ (cid:20)Z r e a ψ ( N − (1 + u ) mp d r (cid:21) /p (cid:20)Z r e a (1 − /p ) p ′ ψ ( N − − /p ) p ′ d r (cid:21) /p ′ , Integrating again the last inequality from 0 to r we conclude the proof because − u ( r ) + u (0) ≤ C k e a/p ψ ( N − /p (1 + u ) m k L p r. where C = e − a ( c ) ψ − N ( c ) hR R e a (1 − /p ) ψ ( N − − /p ) d r i /p ′ . Proof of Theorem 1.6 (Power case).
Take p = (2 bt + m − /m . By Lemma 5.5 and usingLemma 4.5, it follows that Z r e a/p ψ ( N − /p (1 + u (0)) m ≤ Z r e a/p ψ ( N − /p (1 + u ( r ) + C C (cid:18) − b t m [2 bt − (cid:19) − / [ tb ] R / (2 bt + b ) r ) m d r. Thus, we have k u k m ≤ m C C (cid:18) − b t m [2 bt − (cid:19) − tb R bt + b + 2 m C m C m m + 1 (cid:18) − b t m [2 bt − (cid:19) − mtb R m bt + b + m +1 . Using the above inequality and taking the limit λ ր λ ∗ , it follows that k u ∗ k m ≤ m C C (cid:18) − b t m [2 bt − (cid:19) − tb R bt + b + 2 m C m C m m + 1 (cid:18) − b t m [2 bt − (cid:19) − mtb R m bt + b + m +1 (cid:3) Existence of nonminimal solutions
Lemma 6.1.
Let u and v a weak solution and a weak supersolution, respectively, of ( P λ ) .(i) If µ ( λ, u ) > , then u ≤ v a.e. in Ω .(ii) If u is a regular solution and if µ ( λ, u ) = 0 , then u = v a.e. in Ω . Proof.
Let θ ∈ [0 ,
1] and 0 ≤ φ ∈ W , (Ω). By convexity of s → f ( s ) we have I θ,φ := Z Ω ( ∇ g ( θu + (1 − θ ) v ) · ∇ g φ + φA · ∇ g ( θu + (1 − θ ) v )) d v g − Z Ω λf ( θu + (1 − θ ) v ) φ d v g ≥ λ Z Ω ( θf ( u ) + (1 − θ ) f ( v ) − f ( θu − (1 − θ ) v )) φ d v g ≥ . LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 24
Since I ,φ = 0, the derivative of I θ,φ at θ = 1 is nonpositive. If µ ( λ, u ) >
0, clearly u ≤ v .We shall prove that this holds true if µ ( λ, u ) ≥
0. In deed, we have Z Ω ( ∇ g ( u − v ) · ∇ g φ + φA · ∇ g ( u − v )) d v g − Z Ω λf ′ ( u )( u − v ) φ d v g = 0 . (6.1)Since I θ,φ ≥ θ ∈ [0 ,
1] and I ,φ = ∂ θ I ,φ = 0 , we have ∂ θθ I ,φ = − Z Ω λf ′′ ( u )( u − v ) φ d v g ≥ . Take φ = ( u − v ) + . We have ( u − v ) + = 0 in Ω and we get R Ω |∇ g ( u − v ) + | d v g = 0. Itfollows that u ≤ v a.e. in Ω as claimed. Now, if µ ,λ ( u ) = 0 let ψ ,λ the first eigenvalue of L u,λ . Observe that ψ ,λ is in the kernel of the linearized operator L u,λ , and (6.1) is valid ifwe replace u − v with u − v − tψ ,λ . We have Z Ω (cid:0) |∇ g ( u − v − tψ ,λ ) + | + ( u − v − tψ ,λ ) + A · ∇ g ( u − v − tψ ,λ ) + (cid:1) d v g − Z Ω λf ( u )(( u − v − tψ ,λ ) + ) d v g = 0 . We claim that if u < v − tφ ,λ on a set Ω ′ of positive measure, then there exists ǫ > u < v − tφ ,λ a.e. in Ω for any t ≤ t < t + ǫ. Since we have a variational characterizationof φ ,λ we get that ( u − v − tφ ,λ ) + = βφ ,λ a.e. in Ω for some β ∈ R . We can find, byassumption, a set Ω ′ ⊂ Ω of positive measure such that u < v − tφ ,λ − δ for δ > ǫ > u < v − tφ ,λ in Ω ′ for any t ≤ t ≤ t + ǫ. Hence βφ ,λ = 0 a.e. in Ω ′ . Since φ ,λ > β = 0 and u < v + tφ ,λ a.e. in Ωfor any t ≤ t ≤ t + ǫ and this finishes the proof of claim. Now, by contradiction, assumethat u is not equal to v a.e. in Ω. Since u ≤ v, we find a set Ω ′ of positive measure sothat u < v in Ω ′ . Applying the above claim with t = 0 we get some ǫ > u < v − tφ ,λ a.e. in Ω for any 0 ≤ t < ǫ. Set now t = sup { t > u < v − tφ ,λ a.e. in Ω } . Clearly u ≤ v − t φ ,λ a.e. in Ω. The claim and maximal property of t imply that necessarily u = v − t φ ,λ a.e. in Ω since (6.1) holds for any 0 ≤ φ ∈ W , (Ω) . Taking φ = v − u andarguing as before we have R Ω |∇ g ( u − v ) | d v g = 0 contradicting the assumption that u < v on a set of positive measure. (cid:3) Proof of Theorem 1.3.
Using Theorem 1.2, we have that u ∗ exists as a classical solution.On the other hand, we have that µ ,λ ∗ ≥ . If we suppose that µ ,λ ∗ >
0, then the ImplicitFunction Theorem could be applied to the operator L u ∗ ,λ ∗ to allow for the continuation ofthe minimal branch λ ր u λ beyond λ ∗ , which is a contradiction. Therefore µ ,λ ∗ = 0. Theuniqueness of u ∗ in the class of weak solutions follows from the Lemma 6.1. (cid:3) Proposition 6.1. If < λ < λ ∗ , the minimal solutions are stable.Proof. Define λ ∗∗ = sup { λ > u λ is a stable solution for ( P λ ) } . Obviously λ ∗∗ satisfies λ ∗∗ ≤ λ ∗ . If λ ∗∗ < λ ∗ , then u λ ∗∗ is a minimal solution of ( P λ ∗∗ ). For λ ≤ λ ∗∗ , we have that lim λ ր λ ∗∗ u λ ≤ u λ ∗∗ . Since u ∗∗ is solution of ( P λ ∗∗ ) and by minimality LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 25 follows that lim λ ր λ ∗∗ u λ = u λ ∗∗ and µ ,λ ∗∗ ≥
0. If we suppose that µ ,λ ∗∗ = 0 , we get that u λ ∗∗ = u λ for any λ ∗∗ < λ < λ ∗ . But this is a contradiction, which proves that λ ∗∗ = λ ∗ . (cid:3) Proposition 6.2.
For each x ∈ Ω , the function λ → u λ ( x ) is differentiable and strictlyincreasing on (0 , λ ∗ ) . Proof.
Since u λ is stable, the linearized operator L u λ ,λ at u λ is invertible for any 0 < λ < λ ∗ .By the Implicit Function Theorem λ → u λ ( x ) is differentiable in λ . By monotonicity, d u λ d λ ( x ) ≥ x ∈ Ω . Finally, by differentiating ( P λ ) with respect to λ we get that d u λ d λ ( x ) >
0, for all x ∈ Ω. (cid:3) It is standard to show the existence of a second branch of solutions near λ ∗ . We makeuse of Mountain Pass Theorem to provide a variational characterization for this solutions.To apply the Mountain Pass Theorem we will need to truncate the singular nonlinearityinto a subcritical case, that is, we consider a regularized C nonlinearity g ǫ ( u ) , < ǫ < g ǫ ( u ) = − u ) if u < − ǫ ǫ − − ǫ ) pǫ + 2 u p pǫ (1 − ǫ ) p − if u ≥ − ǫ and for Gelfand or Power-type g ǫ ( u ) = f ( u ) if u < t − ǫf ( s − ǫ ) − f ′ ( s − ǫ )( s − ǫ ) p + f ′ ( s − ǫ ) u p p ( s − ǫ ) p − if u ≥ t − ǫ where p > N = 1 , < p < ( N + 2) / ( N −
2) if 3 ≤ N ≤ N ∗ . For λ ∈ (0 , λ ∗ ) and A = ∇ g a we associate the elliptic problem ( − div (cid:0) e − a ∇ g u (cid:1) = λe − a g ǫ ( u ) in Ω ,u = 0 on ∂ Ω , ( S λ )We can define a energy functional on W , (Ω) associated to ( S λ ) given by J ǫ,λ ( u ) = 12 Z Ω e − a |∇ g u | d v g − λ Z Ω e − a G ǫ ( u ) d v g , where G ǫ ( u ) = R u −∞ g ǫ ( s ) d s. We can fix 0 < ǫ < −k u ∗ k ∞ for MEMS case or 0 < ǫ < t −k u ∗ k ∞ for Gelfand and Power-type, and observe that for λ close enough to λ ∗ , the minimal solution u λ of ( P λ ) is also a solution of ( S λ ) that satisfies µ ,λ ( − div( e − a ∇ g ) − λg ′ ǫ ( u λ )) > . Lemma 6.2. If ≤ N < N ∗ and if λ is close enough to λ ∗ , then the minimal solution u λ of ( S λ ) is a strict local minimum of J ǫ,λ on W , (Ω) . Proof.
Since µ ,λ (( − div( e − a ∇ g ) − λg ′ ǫ ( u λ )) > u λ < − ǫ , we have the inequality Z Ω e − a |∇ g φ | d v g − λ Z Ω e − a φ (1 − u λ ) d v g ≥ µ ,λ Z Ω φ d v g , LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 26 for any φ ∈ W , (Ω). Now take φ ∈ W , (Ω) ∩ C (Ω) such that u λ + φ ≤ − ǫ and k φ k C ≤ δ λ . Thus we have J ǫ,λ ( u λ + φ ) − J ǫ,λ ( u λ )= 12 Z Ω e − a |∇ g φ | d v g + Z Ω e − a ∇ g u λ · ∇ g φ d v g − λ Z Ω e − a (cid:18) − u λ − φ − − u λ (cid:19) d v g ≥ µ ,λ Z Ω φ d v g − λ k e − a k ∞ Z Ω (cid:18) − u λ − φ − − u λ − φ (1 − u λ ) − φ (1 − u λ ) (cid:19) d v g . For some
C > (cid:12)(cid:12)(cid:12)(cid:12) − u λ − φ − − u λ − φ (1 − u λ ) − φ (1 − u λ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | φ | and this implies J ǫ,λ ( u λ + φ ) − J ǫ,λ ( u λ ) ≥ (cid:16) µ ,λ − Cλ k e − a k ∞ δ λ (cid:17) Z Ω φ d v g > δ λ is small enough. This proves that u λ is a local minimum of J ǫ,λ in the C topology. We can apply Theorem 2.1 of [23] and get that u λ is a local minimum of J ǫ,λ in W , (Ω). For Gelfand and Power cases we take φ ∈ W , (Ω) ∩ C (Ω) such that u λ + φ ≤ t − ǫ and k φ k C ≤ δ λ . With similar arguments we conclude that u λ is a local minimum of J ǫ,λ in W , (Ω). (cid:3) Now we proof the existence of a second solution for ( S λ ). We need a version of mountainpass theorem [1]. Theorem 6.1 (Critical point of Mountain pass type) . Let J be a C functional definedon a Banach space E that satisfies the Palais-Smale condition, that is, any sequence in E such that ( J ( u n )) n is bounded and J ′ ( u n ) → in E ∗ is relatively compact in E . Assumethe following conditions:(i) There exists a neighborhood B of some u in E and a constant σ > such that J ( v ) ≥ J ( u ) + σ for all v ∈ ∂B. (ii) Exists w B such that J ( w ) ≤ J ( u ) . Defining
Γ = { y ∈ C ([0 , , E ) : γ (0) = u, γ (1) = w } then there exists u ∈ E such that J ′ ( u ) = 0 and J ( u ) = c , where c = inf γ ∈ Γ max ≤ t ≤ { J ( γ ( t )) : t ∈ (0 , } . Lemma 6.3.
Assume that { w n } ⊂ W , (Ω) satisfies J ǫ,λ n ( w n ) ≤ C, J ′ ǫ,λ n → in W − , (Ω) , for λ n → λ > . The sequence ( w n ) then admits a convergent subsequence in W , (Ω) . LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 27
Proof.
By (6.3) we have as n → + ∞ Z Ω e − a |∇ g w n | d v g − λ n Z Ω e − a g ǫ ( w n ) w n d v g = o ( k w n k W , ) . We have the inequality θG ǫ ( u ) ≤ ug ǫ ( u ) for u ≥ M ǫ for some M ǫ > θ >
2. We obtain C ≥ Z Ω e − a |∇ g w n | d v g − λ n Z Ω e − a G ǫ ( w n ) d v g = (cid:18) − θ (cid:19) Z Ω e − a |∇ g w n | d v g + λ n Z Ω e − a (cid:18) θ w n g ǫ ( w n ) − G ǫ ( w n ) (cid:19) d v g + o ( k w n k ) ≥ (cid:18) − θ (cid:19) Z Ω e − a |∇ g w n | d v g + o (cid:16) k w n k W , (Ω) (cid:17) − C ǫ . It follows that sup n ∈ N k w n k W , (Ω) < + ∞ . We have the compactness of embedding W , (Ω) ֒ → L p +1 (Ω) and thus, up to a subsequence, w n ⇀ w weakly in W , (Ω) andstrongly in L p +1 (Ω) for some w ∈ W , (Ω) . It follows that Z Ω e − a |∇ g w | d v g = λ Z Ω g ǫ ( w ) w d v g and we deduce that Z Ω e − a |∇ g ( w n − w ) | d v g = Z Ω e − a |∇ g w n | d v g − Z Ω e − a |∇ g w | d v g + o (1)= λ n Z Ω g ǫ ( w n ) w n d v g − λ Z Ω g ǫ ( w ) w d v g + o (1) → . as n → + ∞ , and the lemma is proved. (cid:3) Proof of Theorem 1.4.
We first show that J ǫ,λ has a mountain pass geometry in W , (Ω).Since u λ is a local minimum for J ǫ,λ for λ ր λ ∗ , condition ( i ) of Theorem 6.1 is satisfied.Consider r > B r ⊂ Ω and a cutoff function χ so that χ = 1 on B r and χ = 0outside B r . Let w ǫ = (1 − ǫ ) χ ∈ W , (Ω) . In MEMS case, we have J ǫ,λ ( w ǫ ) ≤ (1 − ǫ ) Z Ω e − a |∇ χ | d v g − λǫ Z B r e − a d v g → −∞ as ǫ → λ bounded away from 0. With a similar argument we can provethe same result for Gelfand and Power cases. Thus we have J ǫ,λ ( u λ ) → J ǫ,λ ∗ ( u λ ∗ ) as λ → λ ∗ we get for ǫ > J ǫ,λ ( w ǫ ) < J ǫ,λ ( u λ ) LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 28 holds for λ close to λ ∗ . It follows by Lemma 6.3 that the functional J ǫ,λ satisfies the Palais-Smale condition on W , (Ω) . We fix ǫ > λ close to λ ∗ we define c ǫ,λ = inf γ ∈ Γ max u ∈ γ J ǫ,λ ( u ) . We can use the mountain pass theorem to get a solution U ǫ,λ of ( S λ ) for λ close to λ ∗ . Asimilar proof as in Lemma 6.1 shows that the convexity of g ǫ ensures that problem ( S λ )has a unique solution at λ = λ ∗ , which is u ∗ . By elliptic regularity theory we get that U ǫ,λ → u ∗ uniformly in C (Ω). Thus U ǫ,λ ≤ t − ǫ for λ close to λ ∗ . Therefore, U ǫ,λ is asecond solution for ( P λ ) bifurcating from u ∗ , that we denote by U λ . Since U λ is a mountainpass solution, U λ is not a minimal solution. Thus U λ is unstable solution of ( P λ ). (cid:3) References [1] A. Ambrosetti, P. Rabinowitz:
Dual variational methods in critical point theory and applications.
J.Functional Analysis (1973), 349–381. 26[2] P. Antonini, D. Mugnai, P. Pucci: Quasilinear elliptic inequalities on complete riemannian manifold.
J. Math. Pures Appl. (2007), 582–600. 9[3] E. Berchio, A. Ferrero, G. Grillo Stability and qualitative properties of radial solutions of the Lane-Emden-Fowler equation on Riemannian models . J. Math. Pure. Appl. , 1–35, 2014. 8[4] H. Berestycki, A. Kiselev, A. Novikov, L. Ryzhik:
The explosion problem in a flow.
J. Anal. Math. (2010), 31–65. 3, 6[5] H. Berestycki, L. Kagan, G. Joulin, G. Sivashinsky:
The effect of stirring on the limits of thermalexplosion.
Combustion Theory and Modelling (1997), 97–112. 7[6] H. Brezis, J. L. Vazquez: Blow-up solutions of some nonlinear elliptic problems.
Rev. Mat. Univ.Complut. Madrid (1997), 443–469. 3, 5, 6[7] D. Castorina, M. Sanch´on: Regularity of stable solutions to semilinear elliptic equations onRiemannian models.
Adv. Nonlinear Anal. (2015), 295–309. 8[8] C. Cowan, N. Ghoussoub: Regularity of the extremal solution in a MEMS model with advection.
Meth. Appl. Anal. (2008), 355–360. 7[9] X. Cabr´e: Regularity of minimizers of semilinear elliptic problems up to dimension 4.
Comm. PureAppl. Math. (2010), 1362–1380. 6[10] X. Cabr´e, M. Sanch´on: Geometric-type Sobolev inequalities and application to the regularity ofminimizers.
J. Funct. Anal. (2013), 303–325. 6[11] C. Cowan: Optimal Hardy inequalities for general elliptic operators with improvements.
Commun.Pure Appl. Anal. (2010), 109–140. 7[12] M. Crandall, P. Rabinowitz: Bifurcation, perturbation of simple eigenvalues, and linearized stability.
Arch. Rational Mech. Anal. (1973), 161–180. 5[13] M. Crandall, P. Rabinowitz: Some continuation and variational methods for positive solutions ofnonlinear elliptic eigenvalue problems.
Arch. Rational Mech. Anal. (1975), 207–218. 5[14] P. Esposito, N. Ghoussoub, Y. Guo: Compactness along the branch of semistable and unstablesolutions for an elliptic problem with a singular nonlinearity.
Comm. Pure Appl. Math. (2007),1731–1768. 6[15] P. Esposito, N. Ghoussoub, Y. Guo: Mathematical analysis of partial differential equations modelingelectrostatic MEMS.
Courant Lecture Notes in Mathematics (2009). 4, 6[16] A. Farina, L. Mari, E. Valdinoci: Splitting theorems, symmetry results and overdeterminated problemsfor Riemannian manifolds.
Comm. Partial Differential Equations , (2013), 1818–1862. 8[17] G. Gilbarg, N. Trudinger: Elliptic partial differential equations of second order.
Second edition,Grundlehren der Mathematischen Wissenschaften, 224, Springer, Berlin-New York, 1983. 4
LLIPTIC PROBLEMS ON RIEMANNIAN MANIFOLDS 29 [18] N. Ghoussoub, Y. Guo:
On the partial differential equations of electrostatic MEMS devices:stationary case.
SIAM J. Math. Anal. (2007), 1423–1449. 5, 6[19] Y. Guo, Z. Pan, M. Ward: Touchdown and pull-in voltage behavior of a mems device with withvarying dielectric properties.
SIAM J. Appl. Math. (2005), 309–338. 6[20] E. Hebey: Nonlinear analysis on manifolds: Sobolev spaces and inequalities.
Courant Lectures Notesin Mathematics , 2000. 4[21] D. Joseph, T. Lundgren: Quasilinear Dirichlet problem driven by positive sources.
Arch. RationalMech. Anal. (1972), 241–269. 5[22] J. Keener, H. Keller: Positive solutions of convex nonlinear eigenvalue problems.
J. DifferentialEquations (1974), 103–125. 5[23] A. Khan, D. Motreanu: Local minimizers versus X-local minimizers.
Optim. Lett. (2013), 1027–1033. 26[24] T. Kura,: The weak Supersolution-Subsolution Method for second order quasilinear elliptic equations.
Hiroshima Math. J. (1989), 1–36. 9[25] X. Luo, D. Ye, F. Zhou: Regularity of the extremal solution for some elliptic problems with singularnonlinearity and advection.
J. Differential Equations (2011), 2082–2099. 7, 8[26] M. do Carmo, C. Xia,
Complete manifolds with non-negative Ricci curvature and the Caffarelli-Kohn-Nirenberg inequalities.
Compos. Math. (2004), 818–826. 7[27] F. Mignot, J-P. Puel,
Sur une classe de problemes non lineaires avec non linearite positive, croissante,convexe.
Comm. Partial Differential Equations (1980), 45–72. 3, 5[28] G. Neved: Regularity of the extremal solution of semilinear elliptic equations.
C. R. Acad. Sci. ParisSr. I Math. (2000), 997–1002. 6[29] J. Pelesko, A. Bernstein:
Modeling MEMS and NEMS.
Chapman Hall and CRC Press, 2008. 6[30] E. Presnov:
Global decomposition of vector field on riemannian manifold along natural coordinates.
Reports on Mathematical Physics (2008), 273–282. 9[31] J. Wei, D. Ye: On MEMS equation with fringing field.
Proc. Amer. Math. Soc. (2010), 1693–1699. 7(J.M. do ´O)
Department of Mathematics, Federal University of Para´ıba58051-900, Jo˜ao Pessoa-PB, Brazil
E-mail address : [email protected] (R. Clemente) Department of Mathematics, Rural Federal University of Pernambuco52171-900, Recife, Pernambuco, Brazil
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