Stationary states of reaction-diffusion and Schrödinger systems with inhomogeneous or controlled diffusion
aa r X i v : . [ m a t h . A P ] S e p Stationary states of reaction-diffusion and Schr¨odingersystems with inhomogeneous or controlled diffusion
Boyan SIRAKOV
PUC-Rio, Departamento de Matem´atica, G´avea, Rio de Janeiro, CEP 22451-900, Brazil
Alexandre MONTARU
Laboratoire de Math´ematiques, Universit´e de Franche-Comt´e, 25030 Besan¸con, France
Abstract . We obtain classification, solvability and nonexistence theorems for positive sta-tionary states of reaction-diffusion and Schr¨odinger systems involving a balance between re-pulsive and attractive terms. This class of systems contains PDE arising in biological modelsof Lotka-Volterra type, in physical models of Bose-Einstein condensates and in models ofchemical reactions. We show, with different proofs, that the results obtained in [ARMA, 213(2014), 129-169] for models with homogeneous diffusion are valid for general heterogeneousmedia, and even for controlled inhomogeneous diffusions.
This paper is a sequel to our recent work [28], in which we studied stationary states ofsystems of reaction-diffusion PDEs or standing waves of coupled Schr¨odinger systems,including as a particular case the important for applications system − ∆ u = u h v p ( a ( x ) v q − c ( x ) u q ) + µ ( x ) (cid:3) in Ω , − ∆ v = v h u p ( b ( x ) u q − d ( x ) v q ) + ν ( x ) (cid:3) in Ω , (1)where Ω ⊆ R n , p ≥ , q > , q ≥ | − p | , (2)and the coefficients a, b, c, d, µ, ν are H¨older continuous functions in Ω, with a, b > , c, d ≥ , ab − cd ≥ . (3)Cases of particular interest are:(i) p = 0, q = 1 – then (1) is a Lotka-Volterra system, a model on biological speciesinteractions;(ii) p = 0, q = 2 – then (1) models phenomena in nonlinear optics and the theoryof Bose-Einstein condensates;(iii) p = 1, q = 1 – in this case (1) is related to a model of chemical reaction. Moregeneral models in chemistry are obtained by varying p and q .We refer to Section 1.3 in [28] for a more detailed discussion on these applications,as well as references. We observe the last condition in (3) means the reaction in thesystem dominates the absorption, so there is no conservation of mass in the time-dependent version of (1). In [28] we studied the classification and non-existence ofpositive solutions of (1) in unbounded domains, as well as the a priori bounds andexistence results which can be inferred in smooth bounded domains.A major difficulty in the study of system (1) is that it is in general neither co-operative (or quasi-monotone), nor variational in the sense that its solutions cannot1e written as critical points of some functional defined on a Banach space. The coreof the new method developed in [28] consists in proving Liouville type theorems fora system of elliptic inequalities satisfied by some auxiliary sub- and super-harmonicfunctions of u, v . This method gives many new results even for systems which happento be variational, such as (i) and (ii) above with a = d .However, the proofs of the results in [28] depend crucially on the fact that thesecond-order elliptic operator in (1) is the Laplacian, that is, in the models aboveonly homogeneous diffusion can be considered. Our main goal here is to remove thishypothesis and show that the main results in [28] are valid for general operators innon-divergence form. It is remarkable that the (necessarily) different proofs we givehere not only permit to generalize but also to shorten some of the proofs from [28].Our arguments will be entirely based on the maximum principle and its consequencesfrom the regularity theory for elliptic equations.In terms of the applications, if the underlying stochastic process X t is not apure diffusion (i.e. a Brownian motion W t ) but rather follows dX t = Σ( X t ) dW t , forsome positive variance matrix Σ, in the corresponding PDE the Laplacian is replacedby tr ( A ( x ) D u ) = P a ij ( x ) ∂ ij u , where the matrix A = Σ T Σ accounts for the spatialinhomogeneity. If the process is also allowed to have drift dX t = b ( X t ) dt + Σ( X t ) dW t ,then we end up with the differential operator with a first order term Lu = tr ( A ( x ) D u ) + b ( x ) .Du = X a ij ( x ) ∂ ij u + X b i ( x ) ∂ i u. (4)Replacing the Laplacian by such operators in the examples (i)-(iii) above meansallowing heterogeneous media, as well as a possibility to consider gradient-dependentequations, i.e. advection in addition to diffusion and reaction-absorption (for (i) and(iii)), or more general derivative Schr¨odinger equations (for (ii), see for instance [8],chapter I.6).Even more generally, an object of intensive study are controlled processes, in which X t follows dX t = b α t ( X t ) dt + σ α t ( X t ) dW t , where α t is an index process correspondingto a choice made in order to maximize or minimize some cost function (see [6], [21]).The PDE operators modeling such processes are suprema or infima of linear operatorsas in (4), with fixed ellipticity constants and bounds for the coefficients. Theseoperators are usually referred to as Hamilton-Jacobi-Bellman (HJB) operators, andare in turn a subclass of the so-called Isaacs min-max operators, basic in game theory.In the following we will consider Isaacs operators as in (5) below, that is, sup-inf overarbitrary index sets of linear operators as in (4) F [ u ] := sup α ∈A inf β ∈B L ( α,β ) u = sup α ∈A inf β ∈B X i,j a ( α,β ) ij ( x ) ∂ ij u + X i b ( α,β ) i ( x ) ∂ i u ; (5)here A , B are arbitrary index sets and the coefficients a ( α,β ) ij , b ( α,β ) i are continuousfunctions. We assume that all eigenvalues of the symmetric matrices A ( α,β ) belongto a fixed interval [ λ, Λ], and that the L ∞ -norms of the vectors b ( α,β ) are bounded by B , for some constants 0 < λ ≤ Λ, B ≥
0. Note this is equivalent to Definition 3.2below. Note also that F is linear and reduces to (4) when |A| = |B| = 1 in (5),and that F is a HJB operator when |A| = 1 or |B| = 1. In the sequel we willwrite F [ u ] = F ( D u, Du, x ) when we want to distinguish the dependence of F in thederivatives of u and x . Writing F ( D u ) will mean F is autonomous and depends onlyon the second derivatives of u . 2hroughout the paper solutions are understood in the viscosity sense. By applyingthe well-known regularity results for viscosity solutions to the systems we consider,we know that their solutions are in C ,γ for some γ >
0, and even in C ,γ provided theoperator F is of HJB type and the coefficients in the equation are H¨older continuous.We observe that viscosity solutions are not an added complication, they provide agood framework in this setting, just like Sobolev spaces do for some divergence-formoperators, even when one knows that any H -solution is classical.Next, we comment on the novelty of our results. First and foremost, they are thefirst of their kind for models like (i)-(iii) above with controlled diffusions , that is, forsystems like (1) where the Laplacian is replaced by a fully nonlinear operator such asa Isaacs (or even HJB) operator.Let us give some more context for each of the models (i), (ii), (iii). Linear operatorsas in (4) were considered for Lotka-Volterra systems in a number of papers; the mostgeneral results available to date as well as references can be found in [19]. Whenreduced to the linear case, the theorems below strengthen the results from [19].The comparison between the case of the Laplacian and more general operatorsis probably most easily made for (ii). For the last ten years there has been a hugeamount of work for systems in the form (ii), and they all assume the differentialoperator is in divergence form, in particular the Laplacian. When reduced to (ii),our main results from [28] completed the previous works on Schr¨odinger systems, byestablishing existence results for the case a, b, c, d ≥
0, which was almost unstudied.On the other hand, Theorem 1.3 below seems to be the first result whatsoever forSchr¨odinger systems governed by inhomogeneous diffusions, i.e. with non-divergenceform linear elliptic operators, independently of the sign of the coefficients a, b, c, d .Finally, for (iii), the results in [28] seem to be the first on their kind. Here weextend these to spatially inhomogeneous and controlled diffusions, which have obviousrelevance in chemistry.We observe that existence and non-existence for fully nonlinear systems with adifferent class of nonlinearities (with cooperative and fully coupled leading terms suchas Lane-Emden type systems, see (20) below) can be found in [32]. We refer to thatpaper, as well to [9] and [23], for more examples and references on problems wherefully nonlinear systems appear. In passing, and in order to provide a quotable source,below we will record several general nonexistence results for systems of Lane-Emdentype, which essentially follow from a number of recent advances in the theory of Isaacsoperators but do not seem to have appeared before. We will use some of these resultsin our proofs.The following classification theorem is our first main result. It represents a strongrigidity property, and states that nonnegative entire solutions ( u, v ) of the system − F ( D u ) = u h v p ( av q − cu q ) (cid:3) − F ( D v ) = v h u p ( bu q − dv q ) (cid:3) (6)can only be semi-trivial (i.e. u ≡ v ≡
0) or have proportional components (i.e. u/v =const). In other words, rather unexpectedly, the existence of a positive entiresolution of (6) is equivalent to the existence of a positive entire solution of a scalarequation. 3 heorem 1.1.
Assume a, b, c, d, p, q are real numbers such that (2) and (3) hold.If u, v are nonnegative functions which satisfy (6) in the whole space R n , theneither u ≡ , or v ≡ , or there exists a real number K > such that u ≡ Kv . A particularly remarkable feature of this result is that it is independent of anynotion of criticality, that is, of how large p , q or n are.Theorem 1.1 reduces the question of existence of positive entire solutions of thesystem (6) to that of the scalar equations − F ( D u ) = 0 and − F ( D u ) = u p + q +1 in R n . (7)We discuss the available nonexistence results for these equations below.Next, we consider the question of existence of positive solutions of (6) in conesof R n . By a cone we mean a set in the form C ω = { x ∈ R n \ { } : x/ | x | ∈ ω } , forsome C -smooth subdomain of the unit sphere ω . Theorem 1.2.
Assume a, b, c, d, p, q are real numbers such that (2) and (3) hold. Let u, v be nonnegative functions which satisfy (6) in a nonempty open cone C ω ⊂ R n ,and are proportional or vanish on ∂ C ω .(a) If u and v are bounded, then u and v are proportional, or u and v vanish.(b) If c = d = 0 then u and v are proportional, or one of them vanishes. This result is independent of how large p and q are, and reduces the system tothe equations (7), set in C ω .Note that results on classification of bounded solutions of fully nonlinear systemsin cones were previously obtained in [32], for a different type of nonlinearities (ofLane-Emden type), and with different proofs. On the other hand, the statement (b)above is to our knowledge the first classification result for unbounded solutions ofsystems with operators in nondivergence form. We also observe this result coversnonlinearities such as uv , u v , which were recently found to play an important rolein some applications, see for instance [7].Finally, we state the existence results in bounded domains which we can obtainas consequences of the previous theorems and well-known techniques from Leray-Schauder-Krasnoselskii degree theory.We introduce the following notation. For any orthogonal matrix Q and any fixed y ∈ Ω we denote with F Q,y the pure second order operator defined by F Q,y ( D u ) := F ( Q T D u Q, , y ) . If F is a linear operator as in (4), then it is easy to see that for each y there exists Q such that F Q,y ( D u ) = ∆ u . If F is rotationally invariant (such as, for instance,an extremal Pucci operator – see Definition 3.1 below), then F Q,y = F . Recall that F ( D u ) is rotationally invariant if it only depends on the eigenvalues of D u .In the following we assume that Ω is a bounded Lipschitz domain such that eachpoint y on the boundary ∂ Ω has a neighbourhood in Ω which is C -diffeomorphic toa neighbourhood of the origin in some closed cone C ω y . Observe if ∂ Ω is C -smooththen every such cone is a half-space. Theorem 1.3.
Let Ω be as stated, and assume the coefficients a, b, c, d, µ, ν, p, q aresuch that (2) and (3) hold and inf Ω ( ab − cd ) > , µ, ν ≤ in Ω . (8)4 ssume in addition that for each y ∈ Ω there exists an orthogonal matrix Q suchthat if u is a bounded nonnegative solution of the equation − F Q,y ( D u ) = u p + q +1 in R n or C ω y , then u ≡ , (9) (see Proposition 1.1 below). Then the system (1), with the Laplacian replaced by anygeneral Isaacs operator as in (5), has a positive solution in Ω with u = v = 0 on ∂ Ω . It is worth observing that we can consider Lipschitz domains almost “free ofcharge”, instead of only smooth domains. This is also a new feature with respect toprevious works on these types of systems, even for systems with Laplacians.The first hypothesis in (8) cannot be weakened, even for simple systems with theLaplacian (see the remark after Corollary 1.4 in [28]). As will be clear from the proof,instead of assuming that µ, ν ≤ µ, ν are smaller thanthe (positive) first semi-eigenvalue of the Pucci maximal operator, or than the firstsemi-eigenvalue of F itself, if F is a HJB operator (Definition 3.3).A discussion of hypothesis (9) is in order.In general, the only non-existence results in unbounded domains for the equationin (9) concern positive supersolutions. Optimal results for nonexistence of positivesupersolutions of (9) in the whole space or in cones are known, [16], [2], [3], [26], [4],and can be expressed in terms of the so-called scaling exponents of the operator F Q,y in (9). These results will be discussed in detail in the next section, we will recordhere their consequences which apply to hypothesis (9).Given an Isaacs operator F ( D u ), we denote with α ∗ ( F ) the ”scaling exponent”of F , defined as the supremum of all positive α such that − F ( D u ) ≥ − α )-homogeneous solution in R n \ { } (if such α exist; if not, then even − F ( D u ) ≥ R n ). Similarly, we denote with α + ( F, C ω ) the supremumof all positive α such that − F ( D u ) ≥ − α )-homogeneous solution in C ω \ { } (such α always exist) See [5] and [4] for more details on these scaling exponents and theway they describe properties of the operators F . Obviously α ∗ ( F Q,y ) ≤ α + ( F Q,y , C ).It is known that α ∗ (∆) = n − α + (∆ , R n + ) = n −
1, and for every Isaacs operator λ Λ ( n − − ≤ α ∗ ( F ) ≤ Λ λ ( n − − , λ Λ n − ≤ α + ( F, R n + ) ≤ Λ λ n − . (10)These bounds are obtained by evaluating the scaling exponents for the Pucci extremaloperators, and first appeared in [16] and [26].The following proposition contains necessary conditions for (9), which are directconsequences from the Liouville results for the equation in (9) in the next section. Proposition 1.1.
The hypothesis (9) in Theorem (1.3) is satisfied if(i) p + q ≤ α + ( F Q,y , C ω y ) , y ∈ ∂ Ω and p + q ≤ { α ∗ ( F Q,y ) , } , y ∈ Ω , or(ii) Ω is a C -domain, F y ( D u ) = F ( D u, , y ) is a rotationally invariant operatorfor each y ∈ Ω , and p + q ≤ { α ∗ ( F y ) , } , or(iii) Ω is a C -domain, F is linear as in (4), and p + q < n − . α ∗ is nonpositive (this is the case for instance for the Pucci maximal operator if Λ λ ≥ n − λ , Λ, n , p , q , such that (9) is satisfied. In the linearcase, Proposition 1.1 (iii), the result is particularly simple to state.The paper is organized as follows. In the next section we state more generalresults which follow from our proofs, and list various non-existence results for ellipticinequalities and systems in unbounded domains. Section 3 contains some easy orwell-known preliminary results; while in Section 4 we prove nonexistence results inunbounded domains for a class of systems that will be derived from the systems weconsider. Section 5 contains the proofs of the classification theorems in unboundeddomains (Theorems 1.1 and 1.2, and their more general versions, Theorems 2.1, 2.5and 2.6) while in Section 6 we give the proof of the existence result, Theorem 1.3(and its more general version, Theorem 2.11). Our proofs yield more general results than the ones stated in the introduction. Welist these more general theorems in this section. We also discuss here Liouville typeresults for scalar inequalities and systems in unbounded domains.The system (6) is included in the class of systems ( −F [ u ] = f ( x, u, v ) , x ∈ Ω −F [ v ] = g ( x, u, v ) , x ∈ Ω , (11)where the main feature of the nonlinearities f and g (or their leading order terms) isthat they satisfy the condition ∃ K > f ( x, u, v ) − Kg ( x, u, v )][ u − Kv ] ≤ u, v ) ∈ R , x ∈ Ω. (12)Indeed, we recall the following result from [28] (Proposition 1.3 in that paper).
Proposition 2.1 ([28]) . If f = u r v p [ av q − cu q ] and g = v r u p [ bu q − dv q ] , where thereal parameters a, b, c, d, p, q, r satisfy a, b > , c, d ≥ , ab ≥ cd, p, r ≥ , q > , q ≥ | p − r | . (13) then f and g satisfy (12), for a unique number K such that a − cK q ≥ , bK q − d ≥ . We assume here that Ω is the cone C ω = { tx, t > , x ∈ ω } , where ω is a C -smoothsubdomain of the unit sphere in R n . The following theorem contains Theorem 1.2. Theorem 2.1.
Let F be an Isaacs operator as in (5) and f and g satisfy (12). Let Ω = C ω and ( u, v ) be a bounded solution of (11) such that u ≡ Kv on ∂ C ω .Then u ≡ Kv in C ω . − F [ u ] = f ( x, u ) ≥ , u > , in C ω . (14)Two types of nonexistence results are available for this equation: general resultsfor supersolutions and more precise results for solutions when the domain is a half-space and the operator F is rotationally invariant. As far as the latter case is con-cerned, it is proved in [17] and Theorem 3.1 in [31] that the nonexistence of solutionsof the equation − F ( D u ) = f ( u ) in R n (and even in R n − ) implies the nonexistenceof positive solutions in a half-space of R n , for every rotationally invariant operator F and locally Lipschitz nonlinearity f (in [31] only Pucci operators were considered,but the proof is the same for every rotationally invariant operator). An extension ofthis result to systems is proved in [32]. Even stronger results are known if the ellipticoperator is the Laplacian, [13].Next, a nearly optimal result for supersolutions can be obtained by combining theresults and methods from the recent papers [3] and [4]. In order to provide a quotablesource, we state a rather general version of this nonexistence theorem. Theorem 2.2.
Let F ( D u ) be an Isaacs operator, and b ≥ . Set F [ u ] := F ( D u ) + b | x | | Du | . Let α + = α + ( F , C ω ) > and α − = α − ( F , C ω ) < be respectively the supremumand the infimum of all α ∈ R such that −F [ u ] ≥ has a positive ( − α ) -homogeneoussupersolution in C ω \ { } (as in Section 3 of [4]). Let γ < and σ − := 1 + 2 − γα − < , σ + := 1 + 2 − γα + > . (15) Assume that the function g : (0 , ∞ ) → (0 , ∞ ) is continuous and satisfies lim inf s ց s − σ + g ( s ) > and lim inf s →∞ s − σ − g ( s ) > , (16) while the continuous function h : C ω \ B R → (0 , ∞ ) is such that for some ω ′ ⊂ ω and c > h ( x ) ≥ c | x | − γ in C ω ′ \ B R . Then for every R > the differential inequality − F [ u ] ≥ h ( x ) g ( u ) in C ω \ B R (17) does not have a positive solution.Proof. This theorem is proved by repeating the proof of Theorem 5.1 in [3], replacingthe functions Ψ + and Ψ − there by the corresponding functions constructed in [4] andby making use of the comparison principle for F .We list several consequences. 7 heorem 2.3. Let F ( D u ) be an Isaacs operator and B ≥ . Let α + = α + ( F, C ω ) , α − = α − ( F, C ω ) , and σ + , σ − be defined as in (15). The differential inequality − F ( D u ) + B | Du | ≥ h ( x ) g ( u ) in C ω \ B R (18) does not have a positive solution, provided g and h are as in the previous theorem,with (16) replaced by the slightly stronger hypothesis: for some ε > s ց s − σ + + ε g ( s ) > and lim inf s →∞ s − σ − − ε g ( s ) > , (19) Proof.
This is a consequence of the previous theorem, if we observe that for every b > R > B ≤ b/ | x | for | x | > R and take b so small that thescaling exponents of F ( D · ) + ( b/ | x | ) | D · | are sufficiently close to α ± ( F, C ω ), and (16)is satisfied.Next, we give two Liouville theorems for systems in cones.The following result, which applies to the systems from the introduction, is anobvious consequence of Theorem 2.1 and Theorem 2.3. Corollary 2.1.
Under the hypothesis of Theorem 2.1, if F [ u ] ≥ F ( D u ) − B | Du | and there exist c , σ > such that for all x ∈ R n + and v ≥ , f ( x, Kv, v ) = c v σ , with σ < α + ( F, C ω ) , then the only nonnegative bounded solution of (11) in C ω is the trivial one. Finally, we record the following Liouville theorem for the so-called fully nonlinearLane-Emden system . More general right-hand sides can be readily studied by thesame argument (given in Section 6 of [3]).
Theorem 2.4.
Let F ( D u ) , F ( D u ) be Isaacs operators with scaling exponents α +1 , α +2 in a cone C ω . Let r, s ≥ . The only nonnegative solution of the system ( − F ( D u ) + B | Du | ≥ v r − F ( D u ) + B | Du | ≥ u s in C ω \ B R (20) is the trivial one, provided rs ≤ or r ) rs − < α +1 or s ) rs − < α +2 . If B = 0 weak inequalities can be allowed in the last hypothesis.Proof. Repeat the proof on page 2041 in [3] where the whole space instead of a conewas considered. Replace the references to Lemma 3.8 there by references to Lemma5.4 in that paper, and as above, replace the functions Ψ + , Ψ − by the more generalfunctions of this type, constructed in [4] for arbitrary Isaacs operator. If B = 0,reason as in the proof of Theorem 2.3 above.Finally, we state a theorem for nonexistence of unbounded solutions in cones forthe type of systems we are mostly interested in this paper, of which Theorem 1.2 (b)is a very particular case. 8 heorem 2.5. Let p, q, r, s ≥ . We assume that f, g satisfy condition (12) for someconstant K > and that, for some c > , f ( x, u, v ) ≥ c u r v p and g ( x, u, v ) ≥ c u q v s for all u, v ≥ and x ∈ C ω . (21) Let ( u, v ) be a nonnegative classical solution of (11) in R n + , such that u = Kv on ∂ C ω .Let F [ u ] = F ( D u ) for some Isaacs operator F .(i) Either u ≤ Kv or u ≥ Kv in R n + .(ii) If r ≤ α + − p α − α + or s ≥ α − − q α + α − , (22) and s ≤ α + − q α − α + or r ≥ α − − p α + α − , (23) then either u ≡ Kv or ( u, v ) is semi-trivial. Here α + = α + ( F, C ω ) > and α − = α − ( F, C ω ) < . Ω = R n For Ω = R n , we focus on the following system ( − F ( D u ) = u r v p [ av q − cu q ] on R n − F ( D v ) = v r u p [ bu q − dv q ] on R n , (24)where we always assume that the real parameters a, b, c, d, p, q, r satisfy the hypothesis(13) of Proposition 2.1 (this hypothesis reduces to (2) and (3) when r = 1).Recalling that α ∗ ( F ) was defined in the previous section, we have the followingresult, in which Theorem 1.1 is contained. Theorem 2.6.
Let F ( D u ) be an Isaacs operator, (13) holds, and K be the numbergiven by Proposition 2.1. Let ( u, v ) be a positive viscosity solution of (24) in R n .i) Assume that α ∗ ( F ) ≤ or ≤ r ≤ α ∗ ( F ) . If p + q < , we assume moreover that u and v are bounded. Then u ≡ Kv .ii) Assume that α ∗ ( F ) ≤ or (cid:18) p ≤ α ∗ ( F ) and c, d > (cid:19) . If q + r ≤ , we assume moreover that u and v are bounded. Then u ≡ Kv . Observe in this theorem there is no restriction on the total degree σ = p + q + r > p and r separately can guarantee that (24) has nononstandard solutions.An easy consequence is the following Liouville type result for the system (24). Theorem 2.7.
Under the hypotheses of the previous theorem, if ab > cd and − F ( D u ) = u p + q +1 has no bounded positive viscosity solution on R n , (25) then (24) has no bounded positive viscosity solution in R n .
9s far as the hypothesis (25) is concerned, we recall the following result from [3].
Theorem 2.8 ([3]) . Let F ( D u ) be an Isaacs operator. The equation − F ( D u ) = u σ has no positive supersolutions in R n (and even in any exterior domain in R n ) provided α ∗ ( F ) ≤ or ≤ σ ≤ α ∗ ( F ) . It is worth observig that it is an outstanding open question whether the ranges of σ for which the equation in (9) (for instance, if F is a Pucci operator) does not admitentire positive supersolutions or entire positive solutions are different. Note this factis well-known for the Laplacian – the equation − ∆ u = u σ does not have positiveentire solutions if and only if σ < ( n + 2) / ( n − σ ≤ n/ ( n − Theorem 2.9.
Let F ( D u ) , F ( D u ) be Isaacs operators with scaling exponents α ∗ , α ∗ in R n . Let r, s ≥ . The only nonnegative solution of the system ( − F ( D u ) ≥ v r − F ( D u ) ≥ u s in R n \ B R (26) is the trivial one, provided α ∗ ≤ , or α ∗ ≤ , or rs ≤ or r ) rs − < α ∗ or s ) rs − < α ∗ . In the last theorem in this section we discuss the classification of nontrivial non-negative solutions of (24).
Theorem 2.10.
Under the hypothesis of Theorem 2.7, if we moreover assume that q + r > , then any nonnegative bounded viscosity solution of (24) is semitrivial, i.e. ( u, v ) = ( C , or ( u, v ) = (0 , C ) with C , C ≥ . Moreover:- If r = 0 , then ( u, v ) = (0 , .- If r > , p = 0 and c > (resp. d > ), then ( u, v ) = (0 , . We consider the following system with general lower order terms −F [ u ] = u r v p (cid:2) a ( x ) v q − c ( x ) u q (cid:3) + h ( x, u, v ) , x ∈ Ω , −F [ v ] = v r u p (cid:2) b ( x ) u q − d ( x ) v q (cid:3) + h ( x, u, v ) , x ∈ Ω ,u = v = 0 , x ∈ ∂ Ω , (27)where F is a general Isaacs operator as in (5) and Ω is a bounded Lipschitz domainsuch that each point y on the boundary ∂ Ω has a neighbourhood in Ω which is C -diffeomorphic to a neighbourhood of the origin in some closed cone C ω y .The following theorem contains Theorem 1.3 as a very particular case.10 heorem 2.11. Let F be an Isaacs operator, and p, r ≥ , q > , q ≥ | p − r | , q + r > . Assume that the system (24) has no bounded positive viscosity solution in R n or in any cone as in the definition of the Lipschitz property of Ω above (sufficientconditions for this are given in the previous subsection).1. Let a, b, c, d ∈ C (Ω) satisfy a, b > , c, d ≥ in Ω and inf x ∈ Ω [ a ( x ) b ( x ) − c ( x ) d ( x )] > . (28) Let h , h ∈ C (Ω × [0 , ∞ ) ) satisfy lim u + v →∞ h i ( x, u, v )( u + v ) σ = 0 , i = 1 , , (29) and let one of the following two sets of assumptions be satisfied: r ≤ , and, setting ¯ m := min { inf x ∈ Ω a ( x ) , inf x ∈ Ω b ( x ) } > , lim inf v →∞ , u/v → h ( x, u, v ) u r v p + q > − ¯ m, lim inf u →∞ , v/u → h ( x, u, v ) v r u p + q > − ¯ m, (30) or m := min { inf x ∈ Ω c ( x ) , inf x ∈ Ω d ( x ) } > , and lim sup u →∞ , v/u → h ( x, u, v ) u r + q v p < m, lim sup v →∞ , u/v → h ( x, u, v ) v r + q u p < m (31) (with uniform limits with respect to x ∈ Ω in (29)– (31)).Then there exists M > such that any positive classical solution ( u, v ) of (55) satisfies sup Ω u ≤ M, sup Ω v ≤ M. (32)
2. Assume that (28)–(30) hold, a, b, c, d, h , h are H¨older continuous and forsome ǫ > x ∈ Ω , u,v> u − h ( x, u, v ) > −∞ , inf x ∈ Ω , u,v> v − h ( x, u, v ) > −∞ , (33)sup x ∈ Ω , u> u − h ( x, u, < λ +1 ( −M + , Ω) , sup x ∈ Ω , v> v − h ( x, , v ) < λ +1 ( −M + , Ω) , (34)sup x ∈ Ω , u,v ∈ (0 ,ǫ ) (max { u, v } ) − h i ( x, u, v ) < λ +1 ( −M + , Ω) , i = 1 , . (35) where M + is the Pucci maximal operator (see Definition 3.1 below).Then there exists a bounded positive classical solution of (55). Remark 2.1. If F is a HJB operator we can replace the first eigenvalue of the Puccioperator by the first eigenvalue of F in the above theorem. For the convenience of the reader, we begin by recalling some definitions.11 efinition 3.1.
Let < λ < Λ . The extremal Pucci operators are defined by M + ( M ) = Λ X µ i > µ i + λ X µ i < µ i = sup λI ≤ A ≤ Λ I tr( AM ) , M − ( M ) = −M + ( − M ) , for any symmetric matrix M ∈ S n , where ( µ i ) i =1 ..n are the eigenvalues of M . Definition 3.2. F is an Isaacs operator if the following conditions are satisfied : • F is uniformly elliptic and Lipschitz: there exist Λ > λ > , B ≥ , such thatfor all symmetric matrices M, N , and all p, q ∈ R n , x ∈ Ω , ( H ) M − ( M − N ) − B | p − q | ≤ F ( M, p, x ) − F ( N, q, x ) ≤ M + ( M − N )+ B | p − q | , • F is 1-homogeneous: for all t ≥ and M ∈ S n , p ∈ R n , x ∈ Ω , we have ( H ) F ( tM, tp, x ) = t F ( M, p, x ) . We will also need the definition of the principal half-eigenvalue of an Isaacs oper-ator. See [1] for more details.
Definition 3.3.
Let Ω be a bounded domain of R n and F be an Isaacs operator. Wedefine the finite real number λ +1 ( −F , Ω) = sup { µ ∈ R , ∃ u ∈ C (Ω) , u > , −F [ u ] ≥ µu in Ω } . We recall all equalities and inequalities in this paper are understood in the vis-cosity sense. For the notion of viscosity solution, we refer the reader to [14], [10].We recall a transitivity result, whose proof is a simple consequence of ( H ) aboveand Lemma 3.2 in [1]. Lemma 3.1.
Let F be an Isaacs operator. Assume that f, g ∈ C (Ω) , u, v ∈ C (Ω) are viscosity solutions in Ω of ( F [ u ] ≥ f, F [ v ] ≤ g. Then w = u − v is a viscosity solution in Ω of M + ( D w ) + B | Dw | ≥ f − g. The next simple lemma is helpful in exploiting condition (12) on the nonlinearities f and g of the system, as shown in the subsequent result. Lemma 3.2.
Assume that F satisfies ( H ) and F [0] = F (0 , , x ) = 0 . If w, h arecontinuous functions such that sign ( w ) =sign ( h ) and w is a viscosity solution in Ω of ( F [ w ] ≥ h F [ − w ] ≥ − h, then | w | is a viscosity solution in Ω of F [ | w | ] ≥ | h | roof. Assume φ ∈ C (Ω) touches by above | w | at x ∈ Ω. If w ( x ) >
0, then h ( x ) ≥ φ touches w by above at x , we have F ( D φ ( x ) , Dφ ( x ) , x ) ≥ h ( x ) = | h ( x ) | . If w ( x ) <
0, then h ( x ) ≤ φ touches − w by above at x , we have F ( D φ ( x ) , Dφ ( x ) , x ) ≥ − h ( x ) = | h ( x ) | . If w ( x ) = 0, then h ( x ) = 0. Moreover, φ ( x ) = 0 and φ ≥ | w | ≥ x ∈ Ωis a minimum point of φ . Hence D φ ( x ) ≥ Dφ ( x ) = 0, from which we deduce F ( D φ ( x ) , Dφ ( x ) , x ) ≥ F (0 , , x ) = 0 = | h ( x ) | . Lemma 3.3.
Assume that F is an Isaacs operator. Let ( u, v ) be a viscosity solutionof (11) in Ω and assume that the nonlinearities f, g ∈ C (Ω) satisfy (12). Then M + ( D | u − Kv | ) + B | D | u − Kv || ≥ | f − Kg | in Ω in the viscosity sense.Proof. Apply Lemma 3.2 to w = u − Kv, and h = Kg ( · , u, v ) − f ( · , u, v ) . Observe that condition (12) means that h w ≥
0. Also, by the continuity of f, g and (12), it is easy to see that for all x ∈ Ω and v ∈ R , we have Kg ( x, Kv, v ) − f ( x, Kv, v ) = 0, hence if w = 0 then h = 0.The last lemma will be useful when considering system (24) on the whole space(since the auxiliary function Z = min( u, Kv ) will play a crucial role in our analysis). Lemma 3.4.
Assume that F is an Isaacs operator. Let Ω be an open set and let u, v, f, g, h ∈ C (Ω) . Assume that u and v are respectively viscosity solutions of −F [ u ] ≥ f and − F [ v ] ≥ g and that ( f ≥ h on { u ≤ v } g ≥ h on { u > v } . (36) Then w := min( u, v ) ∈ C (Ω) is a viscosity solution in Ω of −F [ w ] ≥ h. Proof.
Assume φ ∈ C (Ω) touches by below w at x ∈ Ω. Then φ ≤ u and φ ≤ v . If u ( x ) ≤ v ( x ) then w ( x ) = u ( x ) and φ touches u by below at x , hence − F ( D φ ( x ) , Dφ ( x ) , x ) ≥ f ( x ) ≥ h ( x ) . Similarly, if u ( x ) > v ( x ) then w ( x ) = v ( x ) and φ touches v by below at x , hence − F ( D φ ( x ) , Dφ ( x ) , x ) ≥ g ( x ) ≥ h ( x ) . Lemma 3.5.
Let F ( D u ) be an Isaacs operator. If u is a viscosity solution of F ( D u ) = 0 on R n and u is bounded from below, then u is constant.Proof. The proof is well known and is the same as for the Laplacian. The result isalso included in Theorem 1.7 in [5].
Lemma 3.6.
Let F ( D u ) be an Isaacs operator. Let z a viscosity solution on R n of − F ( D z ) ≥ . i) Assume α ∗ ( F ) ≤ . If z is bounded by below, then z is constant.ii) Assume α ∗ ( F ) > . Then for some m > z ≥ m | x | α ∗ ( F ) for all | x | ≥ . Proof.
The statement in i) follows from Theorem 4.3 in [3], whereas ii) is a particularcase of Lemma 3.8 in [3].
Lemma 3.7.
Let Ω be an domain of R n and let F ( D u ) be an Isaacs operator. Let h ≥ and u ≥ be a viscosity solution in Ω of − F ( D u ) ≥ h. Then, for any compact K ⊂ Ω , there exist γ = γ ( λ, Λ , n ) > , c = c ( λ, Λ , n, K. Ω) > such that inf K u ≥ c (cid:18)Z K h γ (cid:19) γ . Proof.
This is proved in [25]. The result in that paper is stated for a linear operator,but the same proof applies to any uniformly elliptic Isaacs operator.It is well known (see for instance [14], [10, Proposition 2.9], [11, Theorem 3.8])that it is easy to pass to uniform limits with viscosity solutions.
Lemma 3.8.
Let Ω be a domain of R n and F be an Isaacs operator. Let ( u j ) asequence of viscosity solutions of −F [ u j ] = f j ( x ) in Ω where f j ∈ C (Ω) . Assume that u j → u and f j → f locally uniformly in Ω . Then u isa viscosity solution of −F [ u ] = f ( x ) in Ω . We also recall the following solvability result.
Lemma 3.9.
Let F be an Isaacs operator, R > and f ∈ C ( B R ) . Then there existsa viscosity solution u ∈ C ( B R ) of the problem ( −F [ u ] = f ( x ) x ∈ B R ,u = 0 x ∈ ∂B R (37)14 s well as a unique viscosity solution u ∈ C ,α ( B R ) of the problem ( −M + ( D u ) + B | Du | = f ( x ) x ∈ B R ,u = 0 x ∈ ∂B R (38) Proof.
This result is contained in [11], [15]. R n In this subsection, we present two Liouville type results for inequalities on R n . Thefirst one concerns a coercive inequality and extends Lemma 3.4 from [28]. That lemmafor the Laplacian was proved in [28] by using spherical means – a tool obviously notavailable for more general operators. Here we give a different and more general proof,based on estimates from the regularity theory of elliptic equations, such as growthlemmas and Harnack type inequalities. Lemma 4.1.
Let F ( D u ) be an Isaacs operator and w ≥ be a viscosity solution of F ( D w ) ≥ A | x | w p on R n , (39) where p > and A > .i) If w , then w is unbounded.ii) If p > , then w ≡ .Proof. By ( H ) the function w is a viscosity solution on R n of M + ( D w ) ≥ A | x | w p . Proof of i) . Assume w = 0. We define M ( R ) = sup B R w , and will show the existenceof c > R > R ≥ R M (2 R ) ≥ M ( R ) + c, which implies the statement of i).For all x ∈ R n , we define f ( x ) = A | x | w ( x ) p . Since f ≥ f ∈ C ( R n ), thanks to Lemma 3.9, there exists a unique viscositysolution u R > −M − ( D u R ) = f on B R u R = 0 on ∂B R . We define on B R the function v R = M (2 R ) − u R , which satisfies M + ( D v R ) = f on B R v R = M (2 R ) ≥ w on ∂B R .
15y the comparison principle we obtain w ≤ v R on B R , which impliesinf B R u R + M ( R ) ≤ M (2 R ) . Now, we define on B the function ˜ u R ( x ) = u R ( Rx ) , which is a viscosity solution of − M − ( D ˜ u R ) = A R R | x | w ( Rx ) p ≥ ǫ w ( Rx ) p on B , (40)for some ǫ >
0. Since w = 0, there exists R > B w ( R · ) > . Since M + ( D w ) ≥
0, by the local maximum principle applied to w ( R · ) ≥ R > q > C = C ( q ) > R ≥ R , k w ( R · ) k L q ( B ) ≥ C sup B w ( R · ) ≥ C sup B w ( R · ) = C > . However ˜ u R ≥ q > c > B ˜ u R ≥ c ǫ k w ( R · ) p k L q ( B ) . We choose q = q p and obtain a constant c > R ≥ R we haveinf B ˜ u R ≥ c. Since inf B ˜ u R = inf B R u R , this implies M (2 R ) ≥ M ( R ) + c. Proof of ii) . We will show that w is bounded on R n , which implies w = 0, by i).This can be proved similarly to [28], we include the full proof for completeness.As in [29], we define the function w R ∈ C ( B R ) by w R ( x ) = C R α ( R − | x | ) α for all x ∈ B R , where α = p − . It is easy to see, by direct computation, that if
C > x ∈ B R Λ ∆ w R ≤ A | x | w Rp (41)in the classical sense. See for instance the computation on page 15 in [28].Note w R is radial and its first and second radial derivatives are nonnegative. Then w R is convex, so M + ( D w R ) = Λ ∆ w R . Hence M + ( D w R ) ≤ A | x | w Rp . (42)Since w R ( x ) −→ x → ∂B R + ∞ , there exists R ′ < R such that w R ≥ k w k L ∞ ( B R ) on B R \ B R ′ .Assume that sup B R ′ [ w − w R ] >
0. Since w ≤ w R on ∂B R ′ , this supremum is attained at16ome x ∈ B R ′ . On the other hand w R ∈ C ( B R ′ ), so w R is a legitimate test functionfor (39), and by the definition of a viscosity subsolution we get M + ( D w R ( x )) ≥ A | x | w ( x ) p > A | x | w R ( x ) p , a contradiction with (42). This implies that w ≤ w R on B R ′ and then on B R .Now, for any x ∈ R n , we let R → + ∞ , and obtain w ( x ) ≤ lim R →∞ w R ( x ) = C .Hence w is bounded on R n . Lemma 4.2.
Assume that F ( D u ) is an Isaacs operator. Let V ∈ C ( R n ) , V (cid:9) besuch that for all γ > R → + ∞ R n Z B R \ B R V γ > . Assume that α ∗ ( F ) ≤ or ≤ r ≤ α ∗ ( F ) . (43) Let z ≥ be a viscosity solution of − F ( D z ) ≥ V ( x ) z r , x ∈ R n . Then z = 0 . Proof.
The case α ∗ ( F ) ≤ − F ( D z ) ≥ V ≥ , V
0. Hence we cansuppose that α ∗ ( F ) > z
0. Since z ≥ − F ( D z ) ≥
0, by thestrong maximum principle (see [10, Proposition 4.9]) we have z > R n .First, we note that the hypothesis on V implies, for each γ >
0, the existence of R = R ( γ ) > c = c ( γ ) > R ≥ R , Z B \ B V ( Rx ) γ dx ! γ ≥ c . Let γ = γ ( F, n ) > R = R ( γ ). Let R ≥ R . We set z R ( x ) = z ( Rx ) and m ( R ) = inf B R z = inf B z R . It is easy to see that z R is a viscosity solution of − F ( D z R ) ≥ R V ( Rx )( z R ) r , x ∈ R n . By the quantitative strong maximum principle (see Lemma 3.7), there exists
C > B z R ≥ C k R V ( R · ) z Rr k L γ ( B ) ≥ CR m ( R ) r (cid:18)Z B V ( Rx ) γ dx (cid:19) γ . Hence, for all R ≥ R m ( R ) ≥ CR c ( γ ) m ( R ) r . First case . Assume 0 ≤ r ≤
1. For all R ≥ R , since m ( R ) >
0, we have m ( R ) − r ≥ m ( R ) − r ≥ CR c ( γ )because R m ( R ) is nonincreasing. We then obtain a contradiction when R → ∞ . Second case . Assume 1 < r < α ∗ ( F )+2 α ∗ ( F ) , which is equivalent to r − > α ∗ ( F ).From the argument used in the previous case, we deduce that for all R ≥ R m ( R ) ≤ CR − r − , for some C >
0. On the other hand, for any R ≥ m ( R ) ≥ cR − α ∗ ( F ) , for some c > R → + ∞ . Third case . Assume r = α ∗ ( F )+2 α ∗ ( F ) , which is equivalent to r − = α ∗ ( F ) . As aconsequence of this equality, if we set˜ z R = R α ∗ ( F ) z R , and ˜ m ( R ) := inf ∂B R z Φ = inf ∂B ˜ z R Φ , and we can check that cM ≤ ˜ m ( R ) ≤ C. (44)It is easy to see that ˜ z R is a viscosity solution of − F ( D ˜ z R ) ≥ V ( Rx )˜ z rR , x ∈ R n . Hence − F ( D ˜ z R ) ≥ z R − ˜ m ( R )Φ ≥ R n \ B . (45)Since F ( D ( ˜ m ( R )Φ)) = ˜ m ( R ) F ( D Φ) = 0, and F ( D ˜ z R ) ≤ − V ( Rx )˜ z rR , by applying Lemma 3.1 we get −M − (cid:0) D [˜ z R − ˜ m ( R )Φ] (cid:1) = M + (cid:0) D [ ˜ m ( R )Φ − ˜ z R ] (cid:1) ≥ V ( Rx )˜ z rR . We apply the quantitative strong maximum principle (Lemma 3.7) to the operator M − with Ω = B \ B and K = B \ B . We find γ − > c − > B \ B [˜ z R − ˜ m ( R )Φ] ≥ c − Z B \ B V ( Rx ) γ − ˜ z R ( x ) rγ − dx ! γ − . By the [ − α ∗ ( F )]-homogeneity of Φ we have Φ ≥ c on B \ B , which implies, by (44)and (45), that ˜ z R ≥ c on B \ B . B \ B [˜ z R − ˜ m ( R )Φ] ≥ c Z B \ B V ( Rx ) γ − dx ! γ − ≥ c > R large enough, from the hypothesis on V . Therefore, for R large enough and forall | x | = 1 we have ˜ z R (2 x ) ≥ ˜ m ( R )Φ(2 x ) + c , from which we deduce ˜ m (2 R ) ≥ ˜ m ( R ) + c α ∗ ( F ) M , so ˜ m ( R ) is unbounded, a contradiction.The previous theorem can obviously be applied to any constant function V , butalso for any nontrivial nonnegative subsolution, as we show in the following lemma. Lemma 4.3.
Let F ( D u ) be an Isaacs operator. Let V ∈ C ( R n ) , V (cid:9) be such that F ( D V ) ≥ on R n . Then, for any γ > , lim inf R → + ∞ R n Z B R \ B R V γ > . Proof.
This is a consequence of [10, Theorem 4.8 (2)]. Indeed, if we apply the latterto V R = V ( R · ) ≥ R >
0, then for any γ > C = C ( γ ) > R > R n Z B R \ B R V γ ! γ = Z B \ B V ( Rx ) γ dx ! γ ≥ C sup B R \ B R V ≥ C sup ∂B R V = C sup B R V, where the last equality follows from the comparison principle and F ( D V ) ≥ . Butsince V
0, there exists R > B R V > . R n or a cone In this subsection, we fix a cone C ω = { tx, t > , x ∈ ω } , where ω is a C -smoothsubdomain of the unit sphere S . We will use the following notation B + R = B R ∩ C ω , S + R = S R ∩ C ω , and, given an Isaacs operator F ( D u ), we will denote by Ψ ± ∈ C (cid:0) C ω \ { } (cid:1) thepositive solutions of ( − F ( D Ψ ± ) = 0 , in C ω Ψ + = 0 , on ∂ C ω \ { } (46)19ormalized so that Ψ ± ( x ) = 1 for some given point x ∈ C ω , and such thatΨ ± ( x ) > C ω and Ψ ± ( x ) = t α ± Ψ ± ( tx ) for all t > , x ∈ C ω . Here α + = α + ( F, C ω ) > > α − = α − ( F, C ω ) are uniquely determined. For moredetails on these functions and their construction, see [4]. Recall that when F isthe Laplacian and C ω is the half-space { x n > } , we have Ψ + ( x ) = x n | x | − n , andΨ − ( x ) = x n .We next recall the following Phragm`en-Lindelh¨of principle for fully nonlinearequations, which is a particular case of the results in [12] or [4]. Lemma 5.1.
Assume that F is an Isaacs operator. Let w ∈ C ( C ω ) be a boundedviscosity solution of F [ w ] ≥ on C ω (47) w ≤ on ∂ C ω . (48) Then w ≤ in C ω .Proof. This is a special case of [12, Theorems A] or [4, Theorems 1.6-1.7] and theirproofs. Using the notation of [4], we set Ω = Ω ′ = C ω , so D = C ω and we choose D ′ = C ω . Since w is bounded and α − < < α + , then condition (1.12) of [4, Theorem1.7] is clearly satisfied. Proof of Theorem 2.1:
By combining Lemma 3.3 and Lemma 5.1 we get | u − Kv | ≤ C ω , i.e. u ≡ Kv.
We now turn to the proof of Theorem 2.5. The proof of the corresponding result forthe Laplacian in [28] depended heavily on the use of half-spherical means. Obviously,this tool cannot be used for more general operators, so a different proof is needed.Our proof here is shorter and uses the maximum principle and the boundary Harnackinequality.
Proof of Theorem 2.5 (i) . We define the quotient q u ( r ) = inf S + r u Ψ − . Observe that the Hopf lemma applied to −F [ u ] ≥ ∂u∂ν ≥ c r > S + r ∩ ∂ C ω ,while the boundary Lipschitz estimates applied to −F [Ψ − ] = 0 imply ∂ Ψ − ∂ν ≤ C r on S + r ∩ ∂ C ω (here ν denotes the interior normal to ∂ C ω ), therefore q u ( r ) > − = 0 on ∂ C ω and u ≥ u ≥ q ( r )Ψ − on the boundary ∂B + r . By thecomparison principle we have this inequality in B + r . In other words, q u ( r ) = inf B + r u Ψ − .Therefore the function q u is decreasing in r , and hence has a limit as r → ∞ , whichwe denote with L u ≥
0. In particular, we have u ≥ L u Ψ − in C ω .Similarly, we define L v ≥ v ≥ L v Ψ − in C ω . Now, if both L u > L v > −F [ u ] ≥ c (Ψ − ) σ for some σ >
0. This contradicts Theorem 2.2 since Ψ − ≥ c | x | − α − in any propersubcone of C ω .Let us assume L u = 0. We are going to prove that u ≤ Kv . Set w = ( u − Kv ) + .Then we know that −F [ w ] ≤ ≤ − F [ u ] and w ≤ u in C ω . z R be a solution of (cid:26) F [ z R ] = 0 in B + R z R = w on ∂B + R . By the global C ,α -estimates (see for instance Proposition 2.2 in [4]) we see that thesequence z R is locally uniformly bounded as R → ∞ in each fixed compact of C ω .Therefore we can pass to the limit and get a function z such that w ≤ z ≤ u and F ( D z ) = 0 in C ω . We now define Q w ( r ) := sup S + r w Ψ − , and we see that Q w ( r ) is increasing in r , by an argument similar to the one we usedfor q u .On the other hand Q w ( r ) = sup B + r w Ψ − ≤ sup B + r z Ψ − ≤ C inf B + r z Ψ − ≤ C inf B + r u Ψ − = Cq u ( r ) , where we used the boundary Harnack inequality for the functions z et Ψ − – see forinstance Proposition 2.1 in [4]. Now lim r →∞ q u ( r ) = L u = 0 means the nonnegativeincreasing function Q w tends to zero as r → ∞ , that is, Q w ≡ Proof of Theorem 2.5 (ii) . To deduce (ii) from (i) we use exactly the same argu-ment as in the proof of Theorem 2.8 in [28], replacing the reference to Lemma 3.1there by a reference to Theorem 2.2 above. Observe the conditions (2.6) and (2.7) inTheorem 2.8 in [28] are exactly our (22)–(23) with α + = n − α − = − ✷ In this section, we focus on the system ( − F ( D u ) = u r v p [ av q − cu q ] on R n − F ( D v ) = v r u p [ bu q − dv q ] on R n . (49)In our study of (49) we always assume that the real parameters a, b, c, d, p, q, r satisfy a, b > , c, d ≥ , p, r ≥ , q > , q ≥ | p − r | . (50)The last hypothesis provides the following result, proved in the appendix of [28]. Proposition 5.1.
Assume (50).(i) Then the nonlinearities f and g in the system (49) satisfy (12) for some K > .(ii) Assume moreover that ab ≥ cd . Then the number K > is unique. We have K = 1 if and only if a + d = b + c and K > if and only if a + d > b + c . In addition,if ab > cd (resp. ab = cd ), then a − cK q > (resp. = 0 ), bK q − d > (resp. = 0 ). As in [28], in what follows we set Z = min( u, Kv ) , and W = | u − Kv | and we establish a system of elliptic inequalities satisfied by Z and W .21 emma 5.2. Let F ( D u ) be an Isaacs operator. Assume that (13) holds and let ( u, v ) be a positive viscosity solution of (49). Assume that ab ≥ cd .a) The functions Z and W are viscosity solutions on R n of − F ( D Z ) ≥ , M + ( D W ) ≥ . b) If p + q < , suppose in addition that ( u, v ) is bounded. Then Z is a viscositysolution of − F ( D Z ) ≥ CW β Z r in R n , (51) where C > and β := max( p + q, . c) Assume r > p and c, d > . If q + r < , suppose in addition that ( u, v ) isbounded. Then W is a viscosity solution of M + ( D W ) ≥ CZ p W γ in R n , (52) where C > and γ := max( q + r, . d) Assume ab > cd . Then Z is a viscosity solution of − F ( D Z ) ≥ CZ p + q + r in R n . Most of the arguments in the proof of this lemma are similar to [28] (except inc)), thanks to Lemma 3.3 and Lemma 3.4. We sketch the proof here.
Proof. a) By Proposition 5.1, we have a ≥ cK q , bK q ≥ d. (53)Hence, on the set { u ≤ Kv } , we have f ( u, v ) = u r v p [ av q − cu q ] ≥ cu r v p [( Kv ) q − u q ] ≥ { u > Kv } , we have g ( u, v ) ≥ . Now, we apply Lemma 3.3and Lemma 3.4 to u and Kv with h = 0 and deduce a). b) We have x q − y q ≥ C q x q − ( x − y ), for any real x ≥ y ≥
0, where C q = 1 if q ≥ C q = q if 0 < q <
1. Using (53), on the set { u ≤ Kv } we have f ( u, v ) = u r v p [ av q − cu q ] ≥ aK q u r v p [( Kv ) q − u q ] ≥ aC q K u r v p + q − ( Kv − u ) ≥ aC q K p + q u r ( Kv ) p + q − ( Kv − u ) ≥ C Z r W β for some C > p + q ≥
1, we have ( Kv ) p + q − ≥ ( Kv − u ) p + q − ,whereas if p + q <
1, then ( Kv ) p + q − ≥ C for some C >
0, since v is assumedbounded). Similarly, on the set { u > Kv } , we have g ( u, v ) ≥ C Z r W β , for some C > u and Kv with h = CZ r W β where C = min( C , KC )(since − F ( D [ Kv ]) = Kg ( u, v )) and obtain that Z = min( u, Kv ) is a viscosity solu-tion of − F ( D Z ) ≥ CZ r W β . ) Since ab > cd , we know from Proposition 5.1 that for some small ǫ > a ≥ cK q + ǫ , bK q ≥ d + ǫ . Hence, on the set { u ≤ Kv } we have f ( u, v ) = u r v p [ av q − cu q ] ≥ ǫ u r v p + q + c u r v p [( Kv ) q − u q ] ≥ ǫ u r v p + q ≥ ǫK − p − q Z r + p + q . and, similarly, on the set { u > Kv } we have the same inequality for g .We again apply Lemma 3.4 to u and Kv with h = CZ r + p + q , where we set C = ǫ min( K − p − q , K − q − r ), and obtain that Z = min( u, Kv ) is a viscosity solution of − F ( D Z ) ≥ CZ p + q + r . c) Thanks to Lemma 7.1 i) in [28], we know that, since r > p and c, d >
0, wehave for some C > | Kg ( u, v ) − f ( u, v ) | ≥ C u p v p ( u + Kv ) q + r − p − | u − Kv | . We also note that for x, y ≥ x + y >
0, we have xyx + y ≥
12 min( x, y ) . Hence, | Kg ( u, v ) − f ( u, v ) | ≥ C K p (cid:20) u Kvu + Kv (cid:21) p ( u + Kv ) q + r − | u − Kv |≥ C (2 K ) p Z p ( u + Kv ) q + r − | u − Kv | ≥ C Z p W γ , for some C > q + r ≥
1, we have ( u + Kv ) q + r − ≥ | u − Kv | q + r − ,whereas if q + r <
1, we have ( u + Kv ) q + r − ≥ C ′ for some C ′ >
0, since u and v areassumed bounded).Now, thanks to Lemma 3.3, we obtain that in the viscosity sense M + ( D W ) ≥ CZ p W γ in R n . We now can give the following
Proof of Theorem 2.6.
Let ( u, v ) be a positive viscosity solution of (49) in R n . Let Z and W be defined as in the previous lemma. Proof of i)
Assume that W = 0. From Lemma 5.2 b), we know that Z is aviscosity solution of − F ( D Z ) ≥ C V Z r in R n , (54)where C > V = W β , with β := max( p + q, . From Lemma 5.2 a), we knowthat W is a viscosity solution on R n of M + ( D W ) ≥ β ≥ V is aviscosity solution on R n of M + ( D V ) ≥ . Moreover, V (cid:9)
0, hence, by Lemma 4.3, V satisfies the hypothesis of Lemma 4.2.Therefore Z = 0, a contradiction since u, v >
0. Then W = 0, i.e. u = Kv .23 roof of ii) We can assume q + r > . Indeed, if q + r ≤
1, then u, v are bounded and r ≤ q + r ≤ < α ∗ ( F )+2 α ∗ ( F ) so the resultfollows from a), which we already proved.We can also assume that r > p. Indeed, if p + q <
1, then p + q < < q + r so r > p ; if p + q ≥
1, we can assume r > α ∗ ( F )+2 α ∗ ( F ) (or else the result follows from a)), so r > α ∗ ( F ) ≥ p. Since Z is a viscosity solution on R n of − F ( D Z ) ≥ , by Lemma 3.6 ii) thereexists m > Z p ≥ m p | x | p α ∗ ( F ) for all x ∈ R n \ B . Therefore since p ≤ α ∗ ( F ) , there exists A > Z p ≥ A | x | for all x ∈ R n \ B . Moreover, Z p > B so there exists A > Z p ≥ A | x | for all x ∈ B .Therefore, since r > p and c, d >
0, by Lemma 5.2 c) W is a viscosity solution of M + ( D W ) ≥ A | x | W γ in R n , for some A > γ = q + r >
1. Then, by Lemma 4.1, we have W = 0 . Proof of Theorem 2.7.
Let ( u, v ) be a bounded positive viscosity solution on R n of(49). Then from Theorem 2.6 we know that u = Kv.
Hence v is a solution of − F ( D v ) = K p ( bK q − d ) v σ . But since ab > cd , we know from Proposition 5.1 that bK q − d >
0, so, by using thescaling of the equation and the hypothesis, we get v = 0 and hence u = 0. Proof of Theorem 2.10. . Let ( u, v ) be a nonnegative bounded viscosity solution of(49). First note that u is a viscosity solution of − F ( D u ) + Cu ≥ C = c v p u q + r − ≥
0. We can apply the strong minimum principle and deducethat u = 0 or u >
0. The same is true for v . By Theorem 2.7 at least one of u , v vanishes.Assume for instance that u = 0. First, if p > − F ( D v ) = 0, so byLemma 3.5 we have v = C ≥
0. Second, if p = 0 then F ( D v ) = d v q + r . Now, if d = 0, then v = C , while if d >
0, then v = 0 by Lemma 4.1 because q + r > v = 0. Hence, in all cases ( u, v ) is semitrivial.Finally, if r = 0, then it is clear that if u = 0, then v = 0, and vice versa. The laststatements are obvious. 24 A priori estimates and an existence result in a boundeddomain
In this section we prove Theorem 2.11. We recall that we consider the followingsystem with general lower order terms −F [ u ] = u r v p (cid:2) a ( x ) v q − c ( x ) u q (cid:3) + h ( x, u, v ) , x ∈ Ω , −F [ v ] = v r u p (cid:2) b ( x ) u q − d ( x ) v q (cid:3) + h ( x, u, v ) , x ∈ Ω ,u = v = 0 , x ∈ ∂ Ω , (55)where Ω is a bounded Lipschitz domain of R n , and F satisfies ( H ) − ( H ). Proof of Theorem 2.11.
The proof of this theorem follows the same scheme as theproof of Theorems 6.1 and 6.2 of [28], where F was assumed to be the Laplacian. Wewill not repeat the parts of the proof where it is similar to the one in the previouspaper, and will only highlight the differences (most of which appear after the definitionof the function S below).We consider the following parametrized version of system (55), −F [ u ] = F ( t, x, u, v ) , x ∈ Ω , −F [ v ] = G ( t, x, u, v ) , x ∈ Ω ,u = v = 0 , x ∈ ∂ Ω , (56)where F and G are defined as in [28].We perform exactly the same “blow-up” change of variables as in [28]. The onlydifference is that now the modified functions ˜ u j , ˜ v j satisfy a system where appearsthe elliptic operator F ( D ˜ u j , λ j D ˜ u j , x j + λ j y ) , and λ j →
0. By using the global C ,α -estimates for Isaacs operators and Ascoli’stheorem, we can extract a subsequence of (˜ u j , ˜ v j ) which converges in C (the way toperform such a limit argument for fully nonlinear operators is described in extenso in[32]). After the passage to the limit by Lemma 3.8, we obtain a bounded nonnegativeviscosity solution ( U, V ) of ( − F ( D U, , x ) = U r V p (cid:2) a V q − c U q (cid:3) , − F ( D V, , x ) = V r U p (cid:2) b U q − d V q (cid:3) , (57)either in R n or in a cone of R n which resembles locally the boundary of Ω at aneighborhood of some point of ∂ Ω. In addition c d < a b in view of (28), and U (0) ≥ − α >
0. By the strong maximum principle
U > U and V areproportional, hence both U and V are positive. This contradicts the assumption ofTheorem 2.11.So we can assume (57) is set in R n . By performing a rotation in R n we mayassume the operator in the left-hand side of (57) is F Q,x , for any initially chosenorthogonal matrix Q .By the assumption of the theorem U and V cannot be both positive, so by The-orem 2.10 there is a constant ¯ C > U ≡ ¯ C and V ≡
0. Hencelim j →∞ (˜ u j , ˜ v j ) = ( ¯ C,
0) locally uniformly on R n . (58)25ext, the exclusion of such semi-trivial rescaling limits is done exactly as in [28] ifwe observe that the definition of the principal eigenvalue of a fully nonlinear operator(recall Definition 3.3) trivially implies λ +1 ( −F , B R ) = λ +1 ( −F , B ) R . Thus the proof of the a priori bound (32) is concluded.The proof of the existence part of Theorem 2.11 goes again like in [28], replacingthe Laplacian and its first eigenvalue by the fully nonlinear operator F Q,y and its firsteigenvalue defined in Definition 3.3, also observing that, by [1] (see the discussion afterCorollary 3.6 in that paper) that λ +1 ( −M + ) ≤ λ +1 ( −F ) = sup { λ, −F + λ satisfies the maximum principle } . (59)We use the same fixed point theorem of Krasnoselskii and Benjamin as in [28]and the proof stays almost identical until the definition of the function S . Instead ofsetting S = √ uv as in [28], we now set S = min { u, v } . and we prove that, in the viscosity sense, − F [ S ] ≥ ( A − C ) S in Ω . (60)This implies that λ +1 ( − F, Ω) ≥ A − C , which is in contradiction with the arbitrarychoice of A (see (6.12), (6.13), (6.24) in [28] for more on the choice of A ).We know that, in the viscosity sense, −F [ u ] ≥ u r v p (cid:2) ( a ( x ) + A ) v q − c ( x ) u q (cid:3) + ( A − C ) u + A in Ω , −F [ v ] ≥ v r u p (cid:2) ( b ( x ) + A ) u q − d ( x ) v q (cid:3) + ( A − C ) v + A in Ω . Since we choose A such that A > max {k c k ∞ , k d k ∞ } , we have Av q ≥ c ( x ) u q inthe subset of Ω where v ≥ u , and Au q ≥ d ( x ) v q in the subset of Ω where u ≥ v .Therefore −F [ u ] ≥ ( A − C ) u = ( A − C ) S in Ω ∩ { v ≥ u } , −F [ v ] ≥ ( A − C ) v = ( A − C ) S in Ω ∩ { u ≥ v } . By Lemma 3.4 we get (60).Finally, at the end of the proof, in order to show that the first hypothesis ofthe fixed point theorem (see Theorem 6.3 in [28]) is verified, we again argue bycontradiction. Now, for any (small) δ > u, v )with k ( u, v ) k ≤ δ , of (55) with the right-hand side of this system multiplied by some η ∈ [0 , λ = λ +1 ( −F , Ω) and for some ǫ > −F [ u ] ≤ Cu r v p + q + ( λ − ǫ ) max { u, v } ≤ C (max { u, v } ) σ + ( λ − ǫ ) max { u, v } , −F [ v ] ≤ Cv r u p + q + ( λ − ǫ ) max { u, v } ≤ C (max { u, v } ) σ + ( λ − ǫ ) max { u, v } . As in Lemma 3.4 the maximum of subsolutions is a viscosity subsolution, so −F [max { u, v } ] ≤ C (max { u, v } ) σ + ( λ − ǫ ) max { u, v }≤ ( Cδ σ − + λ − ǫ ) max { u, v } in ΩNow, if we choose δ sufficiently small, by (59) we get max { u, v } ≤
0, a contradiction.Theorem 2.11 is proved. 26 eferences [1] S. Armstrong, Principal eigenvalues and an anti-maximum principle for homo-geneous fully nonlinear elliptic equations. J. Diff. Eq. 246 (7) (2009), 2958–2987.[2] S. Armstrong, B. Sirakov, Sharp Liouville results for fully nonlinear equationswith power-growth nonlinearities. Ann. Sc. Norm. Pisa 10 (3) (2011), 711–728.[3] S. Armstrong, B. Sirakov, Nonexistence of positive supersolutions of elliptic equa-tions via the maximum principle. Comm. Part. Diff. Eq. 36 (2011), 2011–2047.[4] S. Armstrong, B. Sirakov, C. Smart, Singular solutions of fully nonlinear ellipticequations and applications. Arch. Rat. Mech. Anal. 205 (2) (2012), 345–394.[5] S. Armstrong, B. Sirakov, C. Smart, Fundamental solutions of homogeneous fullynonlinear elliptic equations. Comm. Pure Appl. Math. 64 (6) (2011), 737–777.[6] M. Bardi, M. Crandall, L. Evans, H.M. Soner, P. Souganidis, Viscosity solutionsand applications. Lecture Notes in Mathematics, 1660. Fond. C.I.M.E (1997).[7] H. Berestycki, S. Terracini, K. Wang, J-C. Wei, On Entire Solutions of an EllipticSystem Modeling Phase Separations, Adv. Math. 243 (2013), 102–126.[8] J. Bourgain, Global Solutions of Nonlinear Schrdinger Equations, AmericanMathematical Society, Colloquium Publications, Vol. 46 (1999).[9] J. Busca, B. Sirakov, Harnack type estimates for nonlinear elliptic systems andapplications Ann. Inst. H. Poincar´e Nonl. Anal. 21 (5) (2004), 543–590.[10] L. Caffarelli, X. Cabr´e, Fully nonlinear elliptic equations. American Mathemat-ical Society Colloquium Publications, 43, Providence, (1995).[11] L.A. Caffarelli, M.G. Crandall, M.Kocan, A. ´Swiech, On viscosity solutions offully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math.49 (1996), 365–397.[12] I. Capuzzo Dolcetta, A. Vitolo, A qualitative Phragm`en-Lindel¨of theorem forfully nonlinear elliptic equations. J. Diff. Eq. 243 (2) (2007), 578–592.[13] Z. Chen, C.-S. Lin, W. Zou, Monotonicity and nonexistence results to cooperativesystems in the half space, J. Funct. Anal. 266 (2014), 1088–1105.[14] M. Crandall, H. Ishii, P.L. Lions. User’s guide to viscosity solutions of secondorder partial differential equations. Bulletin of the AMS, 27 (1) (1992), 1–67.[15] M.G. Crandall, M. Kocan, P.-L. Lions, A. ´Swiech, Existence results for boundaryproblems for uniformly elliptic and parabolic fully nonlinear equations, Elec. J.Diff. Eq. 24 (1999), 1–20.[16] A. Cutri, F. Leoni, On the Liouville property for fully nonlinear equations, Ann.Inst. H. Poincar´e Anal. Non Lin. 17 (2) (2000), 219–245.[17] E.N. Dancer, Some notes on the method of moving planes. Bull. Austral. Math.Soc. 46 (1992), 425–434. 2718] D. de Figueiredo, P.-L. Lions, R. Nussbaum, A priori estimates and existenceof positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61(1982), 41–63.[19] M. Delgado, J. L´opez-G´omez, A. Su´arez , On the symbiotic Lotka-Volterra modelwith diffusion and transport effects, J. Diff. Eq. 160 (2000), 175–262.[20] P. Felmer, A. Quaas, On critical exponents for the Pucci’s extremal operators,Ann. Inst. H. Poincar´e Nonl. Anal. 20 (5) (2003), 843–865.[21] W.H. Fleming. H. Mete Soner, Controlled Markov Processes and Viscosity So-lutions, Stochastic Modelling and Probability vol. 25, Springer-Verlag, (2005).[22] B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear ellipticequations, Comm. Part. Diff. Eq. 6 (1981), 883–901.[23] H. Ishii, S. Koike, Viscosity solutions for monotone systems of second-order el-liptic PDE’s, Comm. Part. Diff. Eq. 16 (6&7) (1991), 1095–1128.[24] S. Koike, A beginner’s guide to the theory of viscosity solutions. MSJ Memoirs,vol. 13. Mathematical Society of Japan, Tokyo, (2004).[25] N. V. Krylov, Some Lp-estimates for elliptic and parabolic operators with mea-surable coefficients. Discr. Cont. Dyn. Syst. Ser. B 17 (6) (2012), 2073–2090.[26] F. Leoni, Explicit subsolutions and a Liouville theorem for fully nonlinear uni-formly elliptic inequalities in halfspaces. J. Math. Pures Appl. 98 (9) (2012),574–590.[27] F.H. Lin, On the elliptic equation D i [ a ij ( x ) D j U ] − k ( x ) U + K ( x ) U p = 0. Proc.Amer. Math. Soc. 95 (2) (1985), 219–226.[28] A. Montaru, B. Sirakov, P. Souplet, Proportionality of components, Liouvilletheorems and a priori estimates for noncooperative elliptic systems, Arch. Rat.Mech. Anal. 213 (1) (2014), 129–169.[29] Osserman, Robert. On the inequality ∆ uu
0, a contradiction.Theorem 2.11 is proved. 26 eferences [1] S. Armstrong, Principal eigenvalues and an anti-maximum principle for homo-geneous fully nonlinear elliptic equations. J. Diff. Eq. 246 (7) (2009), 2958–2987.[2] S. Armstrong, B. Sirakov, Sharp Liouville results for fully nonlinear equationswith power-growth nonlinearities. Ann. Sc. Norm. Pisa 10 (3) (2011), 711–728.[3] S. Armstrong, B. Sirakov, Nonexistence of positive supersolutions of elliptic equa-tions via the maximum principle. Comm. Part. Diff. Eq. 36 (2011), 2011–2047.[4] S. Armstrong, B. Sirakov, C. Smart, Singular solutions of fully nonlinear ellipticequations and applications. Arch. Rat. Mech. Anal. 205 (2) (2012), 345–394.[5] S. Armstrong, B. Sirakov, C. Smart, Fundamental solutions of homogeneous fullynonlinear elliptic equations. Comm. Pure Appl. Math. 64 (6) (2011), 737–777.[6] M. Bardi, M. Crandall, L. Evans, H.M. Soner, P. Souganidis, Viscosity solutionsand applications. Lecture Notes in Mathematics, 1660. Fond. C.I.M.E (1997).[7] H. Berestycki, S. Terracini, K. Wang, J-C. Wei, On Entire Solutions of an EllipticSystem Modeling Phase Separations, Adv. Math. 243 (2013), 102–126.[8] J. Bourgain, Global Solutions of Nonlinear Schrdinger Equations, AmericanMathematical Society, Colloquium Publications, Vol. 46 (1999).[9] J. Busca, B. Sirakov, Harnack type estimates for nonlinear elliptic systems andapplications Ann. Inst. H. Poincar´e Nonl. Anal. 21 (5) (2004), 543–590.[10] L. Caffarelli, X. Cabr´e, Fully nonlinear elliptic equations. American Mathemat-ical Society Colloquium Publications, 43, Providence, (1995).[11] L.A. Caffarelli, M.G. Crandall, M.Kocan, A. ´Swiech, On viscosity solutions offully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math.49 (1996), 365–397.[12] I. Capuzzo Dolcetta, A. Vitolo, A qualitative Phragm`en-Lindel¨of theorem forfully nonlinear elliptic equations. J. Diff. Eq. 243 (2) (2007), 578–592.[13] Z. Chen, C.-S. Lin, W. Zou, Monotonicity and nonexistence results to cooperativesystems in the half space, J. Funct. Anal. 266 (2014), 1088–1105.[14] M. Crandall, H. Ishii, P.L. Lions. User’s guide to viscosity solutions of secondorder partial differential equations. Bulletin of the AMS, 27 (1) (1992), 1–67.[15] M.G. Crandall, M. Kocan, P.-L. Lions, A. ´Swiech, Existence results for boundaryproblems for uniformly elliptic and parabolic fully nonlinear equations, Elec. J.Diff. Eq. 24 (1999), 1–20.[16] A. Cutri, F. Leoni, On the Liouville property for fully nonlinear equations, Ann.Inst. H. Poincar´e Anal. Non Lin. 17 (2) (2000), 219–245.[17] E.N. Dancer, Some notes on the method of moving planes. Bull. Austral. Math.Soc. 46 (1992), 425–434. 2718] D. de Figueiredo, P.-L. Lions, R. Nussbaum, A priori estimates and existenceof positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61(1982), 41–63.[19] M. Delgado, J. L´opez-G´omez, A. Su´arez , On the symbiotic Lotka-Volterra modelwith diffusion and transport effects, J. Diff. Eq. 160 (2000), 175–262.[20] P. Felmer, A. Quaas, On critical exponents for the Pucci’s extremal operators,Ann. Inst. H. Poincar´e Nonl. Anal. 20 (5) (2003), 843–865.[21] W.H. Fleming. H. Mete Soner, Controlled Markov Processes and Viscosity So-lutions, Stochastic Modelling and Probability vol. 25, Springer-Verlag, (2005).[22] B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear ellipticequations, Comm. Part. Diff. Eq. 6 (1981), 883–901.[23] H. Ishii, S. Koike, Viscosity solutions for monotone systems of second-order el-liptic PDE’s, Comm. Part. Diff. Eq. 16 (6&7) (1991), 1095–1128.[24] S. Koike, A beginner’s guide to the theory of viscosity solutions. MSJ Memoirs,vol. 13. Mathematical Society of Japan, Tokyo, (2004).[25] N. V. Krylov, Some Lp-estimates for elliptic and parabolic operators with mea-surable coefficients. Discr. Cont. Dyn. Syst. Ser. B 17 (6) (2012), 2073–2090.[26] F. Leoni, Explicit subsolutions and a Liouville theorem for fully nonlinear uni-formly elliptic inequalities in halfspaces. J. Math. Pures Appl. 98 (9) (2012),574–590.[27] F.H. Lin, On the elliptic equation D i [ a ij ( x ) D j U ] − k ( x ) U + K ( x ) U p = 0. Proc.Amer. Math. Soc. 95 (2) (1985), 219–226.[28] A. Montaru, B. Sirakov, P. Souplet, Proportionality of components, Liouvilletheorems and a priori estimates for noncooperative elliptic systems, Arch. Rat.Mech. Anal. 213 (1) (2014), 129–169.[29] Osserman, Robert. On the inequality ∆ uu ≥ f ( uu