A remark on natural constraints in variational methods and an application to superlinear Schrödinger systems
aa r X i v : . [ m a t h . A P ] D ec A remark on natural constraints in variationalmethods and an application to superlinearSchr¨odinger systems. ∗ Benedetta Noris and Gianmaria VerziniDecember 14, 2011.
Abstract
For a C -functional J defined on a Hilbert space X , we consider the set N = { x ∈ A : proj V x ∇ J ( x ) = 0 } , where A ⊂ X is open and V x ⊂ X is aclosed linear subspace, possibly depending on x ∈ A . We study sufficientconditions for a constrained critical point of J restricted to N to be afree critical point of J , providing a unified approach to different naturalconstraints known in the literature, such as the Birkhoff-Hestenes naturalisoperimetric conditions and the Nehari manifold. As an application, weprove multiplicity of solutions to a class of superlinear Schr¨odinger systemson singularly perturbed domains. Let X denote a Hilbert space and let J be a functional of class C on X .A natural constraint for J is a manifold N ⊂ X enjoying the property thatevery critical point of J constrained to N is in fact a free critical point. Whensearching for critical points of J , natural constraints are typically used when J does not admit (nontrivial) minima.According to Birkhoff and Hestenes [4], the first example in the literatureappears in a paper by Poincar´e [19] in the study of closed geodesics on a closedconvex analytic surface. Since such a geodesic can not be of minimal length,Poincar´e finds it by minimizing the length functional among paths which satisfythe “natural isoperimetric condition” of dividing the surface into two parts ofequal integral curvature. The aforementioned paper by Birkhoff and Hestenesis the first which considers natural constraints from an abstract point of view.In particular, for the fixed end-point problem in the calculus of variations, theauthors prove that every extremal of the functional is indeed a local minimum ∗ Work partially supported by the PRIN2009 grant “Critical Point The-ory and Perturbative Methods for Nonlinear Differential Equations”.2010
AMS Subject Classification.
Primary 58E05, 35A15; secondary 35J50. h∇ J ( x ) , ξ i i = 0 , where the variations ξ i ’s are related to the second variation of J , and theirnumber is the Morse index of the extremal.In more recent times, apart from natural constraints induced by symmetry[16], the development of this topic followed mainly two directions. On one hand,the ideas of Birkhoff and Hestenes were exploited by Berger [2] in searching forperiodic orbits to Hamiltonian systems, and by Berger and Schechter [3] froma more abstract point of view. In these papers ∇ J has a semilinear structure,while the natural constraint is N B-S = { x ∈ X : h∇ J ( x ) , v i = 0 for every v ∈ V } where V ⊂ X is a closed linear subspace such that J ′′ ( x ) is definite on V forevery x ∈ N B-S . This implies both that N B-S is a manifold and that it is anatural constraint, because for any constrained critical point the correspondingLagrange multiplier is zero. On the other hand, one of the most famous examplesof natural constraint is the so called
Nehari manifold N N = { x ∈ X : x = 0 and G ( x ) := h∇ J ( x ) , x i = 0 } , which is named after the papers by Zeev Nehari [13, 14, 15]. Again, since h∇ G ( x ) , x i = h∇ J ( x ) , x i + J ′′ ( x )[ x, x ] , (1.1)if J ′′ ( x ) along x is non-degenerate for every x ∈ N , then both N is a manifoldand it is a natural constraint (see for instance [1], Proposition 1.4).As we mentioned before, a natural constraint N is particularly useful whensearching for non-minimal critical points, which are minima of the restrictedfunctional, so that one expects to find critical points of J by minimizing J | N .From this point of view, the two types of natural constraints introduced abovebehave in a quite different way. While N B-S is often weakly closed, so that thedirect method of the calculus of variations usually applies, on the other hand N N needs not be, thus exhibiting a lack of compactness. The typical strategyto overcome this difficulty is to provide a sort of “projection” of X \ { } into N N , such as u ¯ t ( u ) u , where ¯ t ( u ) ∈ R is conveniently chosen by studying thecritical points of the function t J ( tu ). In this direction, the main problemsarise when a globally defined projection is not available. An alternative way toproceed is to show that J | N N satisfies the Palais-Smale condition. To this aimit is sufficient to require that the non-degeneracy of J ′′ holds uniformly on N N in the sense thateither J ′′ ( x )[ x, x ] ≥ δ k x k or J ′′ ( x )[ x, x ] ≤ − δ k x k , (1.2)for some δ >
0, for every x ∈ N N . Indeed, under such assumption, one canprove that constrained Palais-Smale sequences are free ones. This allows to2ecover compactness by assuming the usual Palais-Smale condition on J (seefor instance [11]).In the literature it is possible to find a number of generalizations of theabove ideas: when searching for points such that ∇ J ( x ) = 0, one imposes as apreliminary condition the vanishing of the projection of ∇ J ( x ) on some closedsubspace V x ⊂ X , possibly dependent on x . This is the case of the classicalNehari manifold, since h∇ J ( x ) , x i = 0 ⇐⇒ proj span { x } ∇ J ( x ) = 0 . Among others, we wish to mention [25, 18, 21, 17, 20, 12, 23, 24].The main aim of the present paper is to provide conditions in order to extendthe above scheme to the constraint N = { x ∈ A : G ( x ) := proj V x ∇ J ( x ) = 0 } , where A ⊂ X is open and V x ⊂ X is a closed linear subspace, for every x ∈ A .Referring to (1.1) and (1.2), the main feature we want to preserve is that thedifferential of G restricted to V x consists of two terms, one of which vanishes on N , and the other one is a quadratic form related to J ′′ , enjoying some coercivityproperty. It will come out that, apart from some regularity conditions, we willneed two main properties, namely that • V x is invariant under differentiation , in the sense that the differential ofany regular vector field laying in V x for every x maps V x into itself; • V x splits into two subspaces V ± x , with the property that J ′′ ( x ) is coercive/anticoercive on V ± x respectively.We stress the fact that, with respect to the previous literature, we do not require J ′′ to be definite on V x ; this allows a better localization of the critical points,as we show in our application to nonlinear Schr¨odinger systems. To express thedependence of V x on x , it is useful to introduce a vector bundle structure on V , the disjoint union of V x . To do that, we denote by T A the (trivial) tangentbundle of A . In the following we are interested only in trivial C -subbundlesof T A , that is bundles V → A , with V ⊂ T A , equipped with a global C -trivialization τ : V → A × V , for some Hilbert space V . With this notation, V x is the fiber of V at x whichis isomorphic to V via τ x := τ ( x, · ). We observe that, by means of the naturalimmersion which we will systematically omit, τ − can be naturally interpretedas a C map τ − : A × V → A × X. With this notation, the regularity assumptions we mentioned above concern ∂ x τ − , besides J ′ and J ′′ . Our main result is the following.3 Let X be a Hilbert space and let J ∈ C ( X, R ) . For A ⊂ X open, let V ± be two trivial C -subbundles of T A , with fibers V ± and trivializa-tions τ ± respectively, which we assume to induce isometries τ ± x on every fiber.Suppose that V + x ∩ V − x = { } and that V := V + + V − is such that V x is a propersubspace of T x A . Set N := (cid:8) x ∈ A : proj V x ∇ J ( x ) = 0 (cid:9) and assume that there exists δ > such that, for every x ∈ N , it holds(inv) ξ ′ ( x )[ v ] ∈ V x for every ( · , ξ ( · )) C -section of V , v ∈ V x ;(coe) ± J ′′ ( x )[ v, v ] ≥ δ k v k X for every v ∈ V ± x .Furthermore, assume that J ′ ( x ) ∈ X ∗ , J ′′ ( x ) : X × X → R , and ∂ x ( τ ± x ) − : V ± × X → X are bounded as linear/bilinear maps, uniformly for x in N .Then N is a natural constraint for J , and every constrained Palais-Smalesequence for J is indeed a free one. We stress the fact that the above theorem can be exploited in order to obtainthe existence of critical points for J , without the need of defining any globalprojection of A onto N .To better clarify the assumptions above, one can consider the particularcase (which includes most applications) in which V x is constant, except for afinite dimensional subspace. That is, let us consider a fixed closed subspace W ⊂ X , and ξ i ∈ C ( A, X ), i = 1 , . . . , k , an orthonormal set, with W ∩ span { ξ ( x ) , . . . , ξ k ( x ) } = { } and let us set V x = W ⊕ span { ξ ( x ) , . . . , ξ k ( x ) } . In such a situation, V x induces a C -subbundle of T A with fiber V ∼ = W × R k and trivialization τ x v = (proj W v, h v, ξ ( x ) i , . . . , h v, ξ k ( x ) i ) . As a consequence, τ x is trivially an isometry and ∂ x ( τ x ) − : (( w, t , . . . , t k ) , u ) k X i =1 t i ξ ′ i ( x )[ u ]is uniformly bounded as a bilinear map on V × X whenever the linear operators ξ ′ i ( x ) : X → X are. Finally, assumption (inv) can be more explicitly written as ξ ′ i ( x )[ v ] ∈ V x for every v ∈ V x , ≤ i ≤ k. One of the main advantages of the method of natural constraints with respectto other variational methods, such as mountain pass or linking theorems, is thatit allows to better localize the critical points, thus providing a deeper qualitativedescription. This is particularly advantageous when facing multiplicity issues.4o illustrate this point, in the second part of the paper we apply the above resultin order to prove multiplicity of solutions to a class of elliptic systems of gradienttype with superlinear nonlinearities, in singularly perturbed domains. Despitethe fact that we can deal with more general situations, in this introduction wedescribe our results in the case of cubic nonlinearities in a smooth boundeddomain of R N , with N = 2 ,
3. Such type of nonlinearities have been extensivelystudied in the recent years, due to their applications both to nonlinear opticsand to Bose-Einstein condensation. Let us consider the system − ∆ u i = µ i u i + u i X j = i β ij u j , u i > , u i ∈ H (Ω) , i = 1 , . . . , k, (1.3)where µ i > β ij = β ji ∈ R , for every i, j . At least in some particular cases,system (1.3) is well known to admit positive solutions with minimal energy, seefor instance [7, 8, 11]. We aim at extending to (1.3) the results first obtainedby Dancer in the case of a single equation, concerning the effect of the domainshape on the multiplicity of solutions, see [9, 10]. While in these papers thetools are mainly topological, a variational approach to the single equation casehas been introduced by Beyon in [5]. We prove the following. (1.2) Theorem. Let Ω and Ω l , l = 1 , . . . , n , be bounded regular domains suchthat Ω l ∩ Ω m = ∅ for every l = m, Ω \ D = n [ l =1 Ω l , where D is a bounded regular open set which is sufficiently small in a suitablesense. If β ij ≤ ¯ β for every i = j , with ¯ β > sufficiently small, then system (1.3) admits at least (2 n − k positive solutions. We distinguish the solutions because, using suitable natural constraints, wecan prescribe whether u i | Ω l is either large or small, for every i = 1 , . . . , k , l = 1 , . . . , n . Note that, in particular, our result holds true in the purely com-petitive case, i.e. β ij <
0. The smallness of D will be made precise by suitableassumptions in the following; for instance, the result holds if D can be decom-posed in a finite number of parts, each of which lies between two hyperplanessufficiently close. We wish to mention that related systems in similar domainswere considered, from a different point of view, in [6]. Notations.
Given I ∈ C k ( X, Y ), k ≥
1, with X and Y Hilbert spaces,and x , u ∈ X , we write I ′ ( x )[ u ] ∈ Y to denote the (first) differential of I evaluated at x along u . Analogously, I ′′ ( x )[ u, v ] ∈ Y will denote the (bilinearform associated to the) second differential along ( u, v ) ∈ X × X . In case Y = R ,a sequence { x n } n ⊂ X is a Palais-Smale (PS) sequence for I (at level c ) if I ( x n ) → c and I ′ ( x n ) → X ∗ .I satisfies the PS-condition (at level c ) if every PS-sequence admits a convergingsubsequence. We say that a subspace V ⊂ X is proper if V = { } and V = X .The orthogonal projection of a vector u ∈ X on V will be denoted by proj V u .5or Ω ⊂ R N smooth bounded domain, Γ ⊂ ∂ Ω relatively open and p ≤ ∗ :=2 N/ ( N −
2) we denote by C S (Ω , p ) (resp. C S (Ω , Γ , p )) the Sobolev constantrelated to the embedding of H (Ω) (resp. H , Γ (Ω)) into L p (Ω). Finally, wedenote by C any constant we need not to specify. Let X , Y be Hilbert spaces, A ⊂ X open and G ∈ C ( A, Y ). We denote by N the zero set of G , that is N := { x ∈ A : G ( x ) = 0 } . Let us recall a well known condition which ensures that N is a manifold. (2.1) Proposition. Let G ∈ C ( A, Y ) . If, for every x ∈ N , G ′ ( x ) is surjectiveand ker( G ′ ( x )) is a proper subspace of X , then N is a C -manifold and thetangent space to N at x is ker( G ′ ( x )) .Sketch of the proof. Being ker( G ′ ( x )) a closed and proper linear subspace, wehave the nontrivial splitting X = ker( G ′ ( x )) ⊕ ker( G ′ ( x )) ⊥ . Now, since G ′ ( x ) :ker( G ′ ( x )) ⊥ → Y is bijective, the implicit function theorem applies and thisprovides a local parametrization of N around x .Our first aim is to establish some general conditions under which N is anatural constraint for a functional J defined on X . (2.2) Proposition. Let J ∈ C ( X, R ) , G ∈ C ( A, Y ) and let N be defined asabove. Let us assume thatfor every x ∈ N there exists a closed and proper linear subspace V x ⊂ X such that J ′ ( x ) | V x is identically zero; (2.4) G ′ ( x ) | V x is surjective onto Y. (2.5) Then N is a manifold and a natural constraint for J .Proof. Let us first show that ker( G ′ ( x )) is a proper subspace of X , so that, by theprevious proposition, N is a manifold. Clearly ker( G ′ ( x )) can not be the entirespace, by (2.5). Let 0 = v ∈ V ⊥ x (which exists since V x is proper). By (2.5)there exists v ∈ V x such that G ′ ( x )[ v ] = G ′ ( x )[ v ], hence v − v ∈ ker( G ′ ( x )).We turn to the second part of the statement. Let x ∈ N be a critical point of J constrained to N . Then there exists a Lagrange multiplier λ ∈ Y ∗ such that J ′ ( x )[ x ] = λ [ G ′ ( x )[ x ]] for every x ∈ X.
6n particular we have λ [ G ′ ( x )[ v ]] = J ′ ( x )[ v ] = 0 for every v ∈ V x , and being G ′ ( x ) surjective on V x we deduce that λ ≡
0, i.e. x is a free criticalpoint of J .When searching for critical points of J , a typical strategy consists in selectinga candidate critical value via some variational principle, and then to exploit somecompactness, usually in the form of a Palais-Smale condition. Since on naturalconstraints free critical points coincide with constrained ones, it is natural towonder if a similar equivalence holds for Palais-Smale sequences too. It comesout that, in our setting, while the first property depends on the surjectivity of G ′ | V , the latter one leans on the uniform injectivity of the same operator. (2.3) Proposition. Under the assumptions of Proposition 2.2, let us assumemoreover that there exist positive constants ρ , ρ ′ , such that k G ′ ( x )[ v ] k Y ≥ ρ k v k X for every x ∈ N , v ∈ V x , (2.6) k G ′ ( x )[ u ] k Y ≤ ρ ′ k u k X for every x ∈ N , u ∈ X. (2.7) Then, for every sequence { ( x n , λ n ) } ⊂ N × Y ∗ , J ′ ( x n ) − λ n ◦ G ′ ( x n ) → in X ∗ = ⇒ J ′ ( x n ) → in X ∗ . Proof.
By definition we havesup u ∈ Xu =0 | J ′ ( x n )[ u ] − λ n [ G ′ ( x n )[ u ]] |k u k X = k J ′ ( x n ) − λ n ◦ G ′ ( x n ) k X ∗ → . Since J ′ ( x n )[ v ] = 0 for every v ∈ V x n , we deduce thatsup v ∈ V xn v =0 | λ n [ G ′ ( x n )[ v ]] |k v k X → . Now, recalling that G ′ ( x n ) restricted to V x n is surjective, we deduce that k λ n k Y ∗ = sup y ∈ Yy =0 | λ n [ y ] |k y k Y = sup v ∈ V xn v =0 | λ n [ G ′ ( x n )[ v ]] |k G ′ ( x n )[ v ] k Y ≤ ρ sup v ∈ V xn v =0 | λ n [ G ′ ( x n )[ v ]] |k v k X → . Finally, the uniform continuity impliessup u ∈ Xu =0 | λ n [ G ′ ( x n )[ u ]] |k u k X ≤ k λ n k Y ∗ sup u ∈ Xu =0 k G ′ ( x n )[ u ] k Y k u k X ≤ ρ ′ k λ n k Y ∗ , which concludes the proof. 7nder standard additional assumptions, the previous result ensures the ex-istence of a critical point of J belonging to N . (2.4) Corollary. In the assumptions of Propositions 2.2 and 2.3, supposemoreover that inf x ∈N \N J ( x ) > inf x ∈N J ( x ) =: c ∈ R and that J satisfies the Palais-Smale condition at level c . Then there exists x ∈ N such that J ( x ) = c and J ′ ( x ) = 0 .Proof. By Ekeland’s variational principle [22] applied to N there exists { x n } ⊂N and { λ n } ⊂ Y ∗ such that J ( x n ) → c and J ′ ( x n ) − λ n ◦ G ′ ( x n ) → X ∗ as n → + ∞ . By the previous proposition J ′ ( x n ) → X ∗ , and the conclusionfollows in a standard way. (2.5) Remark. In order to prove the previous result it suffices to assume con-ditions (2.6) and (2.7) only on minimizing sequences.
A remarkable particular case of the structure just introduced is when the closedlinear subspaces V x depend in a smooth way on x and G ( x ) is the projectionof ∇ J ( x ) on V x . In this case, assumption (2.4) on J ′ is tautologically satisfied,while we will show that assumption (2.6) on G ′ can be expressed in terms of J ′′ , in case the subspaces are invariant under differentiation.In view of the application of Proposition 2.3, we set Y := V + × V − , h y, z i Y := h y + , z + i V + + h y − , z − i V − ,G : A → Y, G ( x ) := ( G + ( x ) , − G − ( x )) , where y = ( y + , y − ), z = ( z + , z − ) and G ± ( x ) := τ ± x proj V ± x ∇ J ( x ) , so that the set N which appears in the statement of Theorem 1.1 is indeedthe null set of G . Notice first that since V ± are C -subbundles of T A , then G ∈ C ( A, Y ). As a consequence we can evaluate G ′ ( x ), first along directionsin V x and next along directions in X . (2.6) Lemma. For every ¯ x ∈ N , ¯ v + ∈ V +¯ x , ¯ w ∈ V ¯ x it holds (cid:10) ( G + ) ′ (¯ x )[ ¯ w ] , τ +¯ x ¯ v + (cid:11) V + = J ′′ (¯ x )[ ¯ w, ¯ v + ] (and an analogous property holds for G − ). roof. Let ξ ( x ) = ( τ + x ) − τ +¯ x ¯ v + . Note that ( · , ξ ( · )) is C -section of V + , i.e. ξ ( x ) ∈ V + x for every x ∈ A , and ξ (¯ x ) = ¯ v + . By definition of projection we havethat h G + ( x ) , τ +¯ x ¯ v + i V + = h proj V + x ∇ J ( x ) , ξ ( x ) i X = h∇ J ( x ) , ξ ( x ) i X . By differentiating the previous expression at ¯ x , along ¯ w ∈ V ¯ x , we obtain (cid:10) ( G + ) ′ (¯ x )[ ¯ w ] , τ +¯ x ¯ v + (cid:11) V + = J ′′ (¯ x )[ ¯ w, ¯ v + ] + h∇ J (¯ x ) , ξ ′ (¯ x )[ ¯ w ] i X , (2.8)where the last term vanishes because of assumption (inv). (2.7) Lemma. There exists a positive constant ρ ′ such that for every x ∈ N and u ∈ X it holds k G ′ ( x )[ u ] k Y ≤ ρ ′ k u k X . Proof.
To start with, we claim that there exists a positive constant ρ ′′ such thatfor every x ∈ N , v + ∈ V + x and u ∈ X it holds (cid:12)(cid:12)(cid:10) ( G + ) ′ ( x )[ u ] , τ + x v + (cid:11) V + (cid:12)(cid:12) ≤ ρ ′′ k v + k X k u k X , and an analogous property holds for G − . Indeed, reasoning as in the previouslemma, we have that (2.8) holds with u ∈ X instead of ¯ w ∈ V ¯ x . The claimfollows, recalling the definition of ξ , by the assumptions of uniform boundednesson J ′ , J ′′ and ∂ x ( τ + x ) − . Now, by isomorphism, vectors g ± ∈ V ± x are uniquelydetermined so that G ′ ( x )[ u ] = τ + x g + + τ − x g − . With this notation we have k G ′ ( x )[ u ] k Y = h ( G + ) ′ ( x )[ u ] , τ + x g + i Y − h ( G + ) ′ ( x )[ u ] , τ − x g − i Y ≤ ρ ′′ k u k X ( k g + k X + k g − k X ) = ρ ′′ k u k X ( k τ + x g + k V + + k τ − x g − k V − ) ≤ √ ρ ′′ k u k X · k τ + x g + + τ − x g − k Y = ρ ′ k u k X · k G ′ ( x )[ u ] k Y . We notice that τ ± induce a global C -trivialization τ : V → A × Y , withfiber τ x : V + + V − → Y, τ x : v + + v − ( τ + x v + , τ − x v − ) . Even though τ x needs not to be an isometry, we have that h τ + x v + , τ − x v − i Y = 0for every v ± ∈ V ± x , and hence k τ x v k Y = k v + k X + k v − k X ≥ k v k X . Proof of Theorem 1.1.
We apply Proposition 2.3 to our context. Assumption(2.4) holds by definition, since J ′ ( x ) identically vanishes along vectors of V x .As it regards (2.5), for fixed x ∈ N , y ∈ Y , we search for w ∈ V x such that G ′ ( x )[ w ] = y . This is equivalent to solving the abstract variational problem a ( w, v ) = h y, τ x v i Y for every v ∈ V x , a ( w, v ) is the following bilinear form on V x a ( w, v ) := h G ′ ( x )[ w ] , τ x v i Y = h ( G + ) ′ ( x )[ w ] , τ + x v + i V + − h ( G − ) ′ ( x )[ w ] , τ − x v − i V − = J ′′ ( x )[ w, v + ] − J ′′ ( x )[ w, v − ](in the last equality we used Lemma 2.6). Such a problem can be easily solvedby applying Lax-Milgram Theorem, since a ( w, v ) is bounded by Lemma 2.7 andit is coercive because a ( v, v ) = J ′′ ( x )[ v + , v + ] − J ′′ ( x )[ v − , v − ] ≥ δ ( k v + k X + k v − k X ) = δ k τ x v k Y ≥ δ k v k X , where we used the fact that J ′′ ( x ) is symmetric and assumption (coe). The lastcalculation also provides the validity of (2.6) as follows k G ′ ( x )[ v ] k Y · k τ x v k Y ≥ a ( v, v ) ≥ δ k τ x v k Y ≥ δ √ k v k X · k τ x v k Y . Finally, (2.7) was proved in Lemma 2.7, so that all the assumptions of Proposi-tion 2.3 hold true.To conclude the section we provide a version of Theorem 1.1 specialized tothe applications we will present next. (2.8) Theorem.
Let X be a Hilbert space, J ∈ C ( X, R ) , V + ⊂ X a fixedclosed linear subspace. We define V + x ≡ V + , V − x := span { ξ ( x ) , . . . , ξ h ( x ) } , V x := V + x ⊕ V − x , with ξ i ∈ C ( A, X ) for every i = 1 , . . . , h , A ⊂ X open, in such a way that V x is proper. As usual, let N := (cid:8) x ∈ A : proj V x ∇ J ( x ) = 0 (cid:9) and c := inf N J. Let us suppose that(i) c ∈ R , inf N \N
J > c ;(ii) J satisfies the PS-condition at level c .Moreover, let us assume that for some < δ < δ ′ there holds, for every x ∈ N with J ( x ) ≤ c + 1 ,(iii) k ξ i ( x ) k X ≥ δ , h ξ i ( x ) , ξ j ( x ) i X = 0 , for every i = j ;(iv) ξ ′ i ( x )[ v ] ∈ V x for every i and v ∈ V x ;(v) ± J ′′ ( x )[ v, v ] ≥ δ k v k X for every v ∈ V ± x ; vi) k ξ ′ i ( x )[ u ] k X ≤ δ ′ k u k X , | J ′ ( x )[ u ] | ≤ δ ′ k u k X and | J ′′ ( x )[ u, w ] | ≤ δ ′ k u k X k w k X for every u, w ∈ X .Then there exists x ∈ N such that J ( x ) = c and J ′ ( x ) = 0 .Proof. First of all, by virtue of Remark 2.5, we can work in the sublevel of J .We choose V + := V + , V − := R h (which have trivial intersection by (v)), and τ + x to be the identity, τ − x : V − x → R h defined as τ − x : ξ (cid:18) h ξ, ξ ( x ) i X k ξ ( x ) k X , . . . , h ξ, ξ h ( x ) i X k ξ h ( x ) k X (cid:19) . In particular, assumption (iii) immediately implies that τ − x is an isometric iso-morphism. Taking into account Theorem 1.1 and Corollary 2.4, the only non-trivial things to check are that assumption (inv) holds and that ∂ x ( τ − x ) − isuniformly bounded as a bilinear map on V − × X . On one hand, if ξ ( x ) = P i t i ( x ) ξ i ( x ), then for any v ∈ V x it holds ξ ′ ( x )[ v ] = h X i =1 t ′ i ( x )[ v ] ξ i ( x ) + h X i =1 t i ( x ) ξ ′ i ( x )[ v ] , where the first term belongs to V − x , while the second one is an element of V x byassumption (iv). On the other hand, if t ∈ R h and u ∈ X , then ∂ x ( τ − x ) − : ( t, u ) h X i =1 t i (cid:18) ξ ′ i ( x )[ u ] k ξ i ( x ) k − h ξ i ( x ) , ξ ′ i ( x )[ u ] i ξ i ( x ) k ξ i ( x ) k (cid:19) , which is uniformly bounded by assumptions (iii) and (vi). In this section we apply Theorem 2.8 in order to obtain multiple positive solu-tions for the system − ∆ u i = ∂ i F ( u , . . . , u k ) , i = 1 , . . . , k, (3.9)where every u i is H on a bounded regular domain Ω ⊂ R N . We stress that, with“positive solutions”, we mean that every component u i must be non negativeand non identically zero. We denote by e , . . . , e k the canonical base of R k , sothat u = ( u , . . . , u k ) = X i u i e i . Throughout this section we will assume that F ∈ C ( R k , R ) and that there exist p ∈ (2 , ∗ ), C F > δ > u, λ ∈ R k , it holds(F1) P i,j | ∂ ij F ( u ) | ≤ C F | u | p − , P i | ∂ i F ( u ) | ≤ C F | u | p − and | F ( u ) | ≤ C F | u | p ;11F2) P i,j ∂ ij F ( u ) λ i u i λ j u j − (1 + δ ) P i ∂ i F ( u ) λ i u i ≥ ∂ i F ( u ) u i ≤ ∂ i F ( u i e i ) u i for every i ;(F4) for every i there exists ¯ u i > ∂ i F (¯ u i e i ) > ∇ F ( u ) · u − (2 + δ ) F ( u ) ≥ t
7→ ∇ F ( tu ) · tu − (2+ δ ) F ( tu ) is nondecreasingfor t ∈ (0 , ∂ i F ( u i e i ) u i ≥ ∂ i F (¯ u i e i )¯ u i ¯ u δi u δi for u i ≥ ¯ u i (3.11)(again by (F2), the function t ∂ i F ( te i ) t/t δ is nondecreasing for t > J ( u ) = 12 Z Ω |∇ u | dx − Z Ω F ( u ) dx. It is standard to prove that J ∈ C ( X, R ) where X := H (Ω , R k ) is endowedwith the norm k u k = R Ω |∇ u | dx = P i R Ω |∇ u i | dx . Since we search forpositive solutions, we assume without loss of generality that F is even withrespect to each component. All solutions will be found as minimizers of J on suitable even constraints. By standard arguments we obtain that, for anyminimizer ( u , . . . , u k ) with u i = 0, then also ( | u | , . . . , | u k | ) is a minimizer,which components are strictly positive by the strong maximum principle. Forthis reason, with a slight abuse, from now on we will work only with k -tupleshaving non-negative components. We start investigating the existence of ground state solutions. Such a problemhas already been successfully faced in [8, 11], nonetheless we prefer to provethe result as a direct application of Theorem 2.8. This will be useful in thefollowing, where we turn to the analysis of excited states. (3.1) Theorem.
Let F ∈ C ( R k , R ) satisfy (F1)-(F4). Then there exists apositive solution of (3.9) in H (Ω) . Before proving this result, we state in the following lemma some preliminaryestimates which will be useful also in the next subsections.12
Let u ∈ X be such that J ′ ( u )[ u i e i ] = 0 for every i = 1 , . . . , k .Then J ( u ) ≥ δ δ k u k . Moreover, denoting by C S (Ω , p ) the Sobolev constant of the embedding H (Ω) ⊂ L p (Ω) , either k u i k ≥ ( C F C S (Ω , p ) p ) − / ( p − or u i ≡ . Proof.
Recalling the definition of J , the assumption writes Z Ω |∇ u i | dx = Z Ω ∂ i F ( u ) u i dx, for every i . As it regards the first part, using equation (3.10) we have that J ( u ) ≥ Z Ω |∇ u | dx −
12 + δ Z Ω ∇ F ( u ) · u dx = δ δ k u k . On the other hand, assumptions (F3) and (F1) give Z Ω |∇ u i | dx = Z Ω ∂ i F ( u ) u i dx ≤ Z Ω ∂ i F ( u i e i ) u i dx ≤ C F Z Ω | u i | p dx ≤ C F C S (Ω , p ) p (cid:18)Z Ω |∇ u i | dx (cid:19) p/ . Proof of Theorem 3.1.
We define A = { u ∈ X : u i i } and V + = { } , ξ i ( u ) = u i e i , i = 1 , . . . , k. Within this setting we have N = (cid:26) u ∈ A : Z Ω |∇ u i | dx = Z Ω ∂ i F ( u ) u i dx, i = 1 , . . . , k (cid:27) , so that Lemma 3.2 holds true for any of its elements. Let us check the assump-tions of Theorem 2.8.(i) The first part of Lemma 3.2 shows that c ≥
0, while the second partimplies that
N \ N = ∅ , thus the only thing to prove is that c < + ∞ , that is N 6 = ∅ . To this aim let u ∈ X be fixed in such a way that u i ≥ u i u i · u j ≡ i = j . We claim that there exists λ ∈ R k , with all positivecomponents, such that ( λ u , . . . , λ k u k ) ∈ N . For each i let us define thesmooth function g i ( λ i ) := λ i Z Ω |∇ u i | dx − Z Ω ∂ i F ( λ i u i e i ) λ i u i dx,
13o that the claim is equivalent to the existence of λ such that g i ( λ i ) = 0 forevery i . On one hand, by assumption (F1) we have g i ( λ i ) ≥ λ i Z Ω |∇ u i | dx − λ pi C F Z Ω u pi dx, which is positive for λ i small. On the other hand, (3.11) implies g i ( λ i ) ≤ λ i Z Ω |∇ u i | dx + C − Z { λ i u i ≥ ¯ u i } ∂ i F ( λ i u i e i ) λ i u i dx ≤ λ i Z Ω |∇ u i | dx + C − Z { λ i u i ≥ ¯ u i } ∂ i F (¯ u i e i )¯ u i ¯ u δi ( λ i u i ) δ dx, which, by (F4), is negative for λ i sufficiently large.(ii) It is a standard consequence of equation (3.10) (see for example [22]).(iii) On one hand it is trivial to check that the ξ i ’s are orthogonal, on theother hand Lemma 3.2 implies that k ξ i ( u ) k ≥ δ > u ∈ N , any vector belonging to V u has the form v = ( λ u , . . . , λ k u k )for some λ ∈ R k . Hence ξ ′ i ( u )[ v ] = λ i u i e i ∈ V u .(v) Let u ∈ N and v = ( λ u , . . . , λ k u k ) ∈ V u . Assumption (F2) and thedefinition of N provide J ′′ ( u )[ v, v ] ≤ X i Z Ω λ i |∇ u i | dx − (1 + δ ) X i Z Ω ∂ i F ( u ) λ i u i dx = − δ k v k . (vi) Using assumption (F1), H¨older inequality and Sobolev embedding wehave, for every v, w ∈ X , and u ∈ N , k ξ ′ i ( u )[ v ] k = k v i k ≤ k v k , | J ′ ( u )[ v ] | ≤ Z Ω |∇ u ||∇ v | dx + C F Z Ω | u | p − | v | dx ≤ (cid:0) k u k + C k u k p − (cid:1) k v k , | J ′′ ( u )[ v, w ] | ≤ Z Ω |∇ v ||∇ w | dx + C F Z Ω | u | p − | v || w | dx ≤ (cid:0) C k u k p − (cid:1) k v kk w k . We can easily conclude observing that, by Lemma 3.2, k u k is uniformly boundedon N ∩ { J ≤ c + 1 } . (3.3) Corollary. For every I ⊂ { , . . . , k } , ˜Ω ⊂ R N smooth and bounded do-main, there exists a (minimal energy) solution of (3.9) in H ( ˜Ω) such that u i > for i ∈ I , u i ≡ otherwise.Proof. It suffices to observe that, letting ˜ k = I and σ : n , . . . , ˜ k o → I increasing, then ˜ F (˜ u , . . . , ˜ u ˜ k ) = F X i ∈ I ˜ u i e σ ( i ) ! satisfies (F1)-(F4) on R ˜ k . 14 Neglecting assumption (F3), it is possible to use the standardNehari manifold in order find nontrivial solutions with possibly vanishing com-ponents. In the setting above, this corresponds to replacing span { u e , . . . , u k e k } with span { u } . Indeed, assumption (F3) is used only in the second part of Lemma3.2, which argument can be directly applied to R Ω |∇ u | dx . The same idea canbe carried on also in the results below. We will prove multiplicity of positive solutions for system (3.9) when Ω is closeto the union of disjoint subdomains. More precisely we introduce the followingnotations and assumptions.(Ω1) Ω and Ω l , l = 1 , . . . , n , are bounded regular domains and D is a boundedregular open set, such thatΩ l ∩ Ω m = ∅ for every l = m, Ω \ D = n [ l =1 Ω l , (Ω2) B ⊃ Ω is a fixed ball, Γ l ( ∂ Ω l , l = 1 , . . . , n , are (non-empty and)relatively open, such that ∂D ∩ Γ l = ∅ (Ω3) η l ∈ C ∞ ( R N ), l = 1 , . . . , n , are such that 0 ≤ η l ≤ η l | Ω l = 1, and η l · η m ≡ l = m . C η > D , η , . . . , η n ) with the property that Z Ω |∇ η l | ϕ dx ≤ C η Z Ω |∇ ϕ | dx for every ϕ ∈ H (Ω) (3.12)(observe that the first integral is actually on D ).In our construction, we assume Ω l , Γ l and B to be fixed, while D , and henceΩ and η l , to vary. From this point of view, since H (Ω) ⊂ H ( B ), the role of B is only to provide Sobolev constants not depending on D , neither on Ω. Aswe mentioned, we consider the case in which D is suitably small, meaning thatboth the Lebesgue measure | D | and the constant C η above are small. This lastproperty is related to the smallness of the N -capacity of suitable subsets of D ,and it can be shown to hold, for instance, if D can be decomposed in a finitenumber of parts, each of which lies between two hyperplanes sufficiently close.We are going to distinguish different solutions of (3.9) by prescribing the“size” of u i | Ω l , for every i = 1 , . . . , k and l = 1 , . . . , n . More precisely let us fixany L i ⊂ { , . . . , n } , L i = ∅ , i = 1 , . . . , k. (3.13)We will provide a solution such that u i | Ω l is “large” for l ∈ L i and “small” for l L i . Due to the arbitrary choice of the sets L i ’s, this will imply the existence15f (2 n − k different positive solutions of system (3.9). The size of each bumpwill be classified in relation to the constants r l := (cid:16) p C F C S (Ω l , Γ l , p ) p (cid:17) − / ( p − , where C S (Ω l , Γ l , p ) is the Sobolev constant of the embedding H , Γ (Ω l ) ⊂ L p (Ω l )(compare with the constant which appears in Lemma 3.2). Let us remark that r l is independent of D . We can finally state the main result of this section. (3.5) Theorem. Let F ∈ C ( R k , R ) satisfy (F1)-(F4) and let Ω ⊂ R N satisfy( Ω Ω | D | , C η are sufficiently small.Then for any L , . . . , L k as in (3.13) there exists a positive solution u of (3.9) such that, for every i and l , Z Ω l |∇ u i | dx > r l for l ∈ L i , Z Ω l |∇ u i | dx < r l for l L i . To start with, using the results of the previous subsection, it is easy toprovide a k -tuple g of non-negative functions in H ( ∪ l Ω l ) such that − ∆ g i = ∂ i F ( g , . . . , g k ) , and g i | Ω l is either positive or zero,depending on whether l ∈ L i or not (in some sense, one can think of g as therequired solution, in the singular limit case D = ∅ ). Indeed, for any l one canapply Corollary 3.3 with ˜Ω = Ω l and I = { i : l ∈ L i } . Then g is the sum of thecorresponding solutions. By trivial extension, g ∈ H (Ω).Let us define the constant (independent of D ) R := max (cid:26) k g k , δδ J ( g ) (cid:27) + 1 , where δ has been introduced in assumption (F2). In order to apply Theorem2.8 we define V + := ( v ∈ H (Ω , R k ) : v i ∈ H Ω \ [ l ∈ L i Ω l ) !) and ξ i,l ( u ) := η l u i e i , i = 1 , . . . , k and l ∈ L i , the latter being smooth on A := ( u ∈ X : k u k < R, R Ω l |∇ u i | dx > r l if l ∈ L i , R Ω l |∇ u i | dx < r l if l L i ) .
16n one hand we have that u i e i = − X l ∈ L i η l ! u i e i + X l ∈ L i η l u i e i ∈ V u , since the first term is in V + and the second one in V − u . This in particularimplies, for every i , u ∈ N = ⇒ J ′ ( u )[ u i e i ] = 0 . (3.14)Analogously, for every i and l , η l u i e i ∈ V u , (3.15)indeed, either it belongs to V + if l L i , or it is equal to ( η l − η l ) u i e i + η l u i e i ,the former belonging to V + and the latter to V − u . We deduce that, for every u ∈ N and l = 1 , . . . , n , it holds0 = J ′ ( u )[ η l u i e i ] = Z Ω ∇ u i · ∇ ( η l u i ) dx − Z Ω ∂ i F ( u ) η l u i dx, which implies Z Ω |∇ ( η l u i ) | dx = Z Ω ∂ i F ( u ) η l u i dx + Z Ω |∇ η l | u i dx. (3.16)Using this property we can prove a result which can be seen as a perturbationof the second part of Lemma 3.2. Such result will allow to better localize thebumps of the elements of N . (3.6) Lemma. Let | D | , C η be sufficiently small. Then there exist positiveconstants C , ε such that, for every u ∈ N , it holds Z Ω l |∇ u i | dx > r l = ⇒ Z Ω l |∇ u i | dx ≥ (1 + C ) r l Z Ω l |∇ u i | dx < r l = ⇒ Z Ω l |∇ u i | dx ≤ ε , where ε can be made arbitrarily small with | D | , C η .Proof. Using (3.16), (F3) and (F1) we have Z Ω l |∇ u i | dx ≤ Z Ω |∇ ( η l u i ) | dx = Z Ω ∂ i F ( u ) η l u i dx + Z Ω |∇ η l | u i dx = Z Ω l ∂ i F ( u ) u i dx + Z D ∂ i F ( u ) η l u i dx + Z Ω |∇ η l | u i dx ≤ Z Ω l ∂ i F ( u i e i ) u i dx + C F Z D | u | p dx + C η R ≤ C F Z Ω l | u i | p dx + C F | D | (2 ∗ − p ) / ∗ C S ( B, ∗ ) p R p + C η R ≤ C F C S (Ω l , Γ l , p ) p (cid:18)Z Ω l |∇ u i | dx (cid:19) p/ + ε ′ , ε ′ denotes a quantity arbitrarily small whenever | D | and C η are. Theconclusion easily follows by observing that, denoting by h ( t ) = C F C S (Ω l , Γ l , p ) p t p − t + ε ′ , it holds h ′ ( r l ) = 0 and h ( r l ) < ε ′ small. End of the proof of Theorem 3.5.
We check the assumptions of Theorem 2.8.(i) To start with, we have that c < + ∞ , since g ∈ N . Secondly, by equation(3.14), we have that Lemma 3.2 holds true also in the present case, thus pro-viding c ≥
0. Finally, let u ∈ N \ N : then, by Lemma 3.6 necessarily k u k = R .But then, using again Lemma 3.2 and the definition of R we obtain J ( u ) ≥ δ δ R > J ( g ) ≥ c. (ii) The same as in the previous subsection.(iii) By definition of A we have that k ξ i,l ( u ) k > r l for every i , l ∈ L i .(iv) It follows from (3.15).(v) If u ∈ N and v ∈ V + then v i ≡ l for every l ∈ L i , whereas for l L i it holds R Ω l |∇ v i | dx < ǫ where ε is defined as in Lemma 3.6. Hence wehave J ′′ ( u )[ v, v ] = Z Ω |∇ v | dx − Z D X i,j ∂ ij F ( u ) v i v j dx − X l L i Z Ω l X i,j ∂ ij F ( u ) v i v j dx ≥ − C F | D | (2 ∗ − p ) / ∗ C S ( B, ∗ ) p R p − − X l L i C F C S (Ω l , Γ l , p ) p ε p − k v k . On the other hand, if v ∈ V − u then v = P i (cid:0)P l ∈ L i t i,l η l (cid:1) u i , for some t i,l ∈ R .Using (F2) and (3.16) we obtain J ′′ ( u )[ v, v ] ≤ k v k − (1 + δ ) Z Ω X i ∂ i F ( u ) X l ∈ L i t i,l η l u i dx = − δ k v k + (1 + δ ) Z Ω X i X l ∈ L i t i,l |∇ η l | u i dx ≤ − δ k v k + (1 + δ ) C η Z Ω X i X l ∈ L i t i,l |∇ u i | dx = − δ k v k + (1 + δ ) C η X i X l ∈ L i R Ω |∇ u i | dx R Ω l |∇ u i | dx Z Ω l t i,l |∇ u i | dx ≤ − δ k v k + (1 + δ ) C η X i X l ∈ L i R r l Z Ω l t i,l |∇ u i | dx ≤ (cid:18) − δ + (1 + δ ) C η R min l ∈ L i r l (cid:19) k v k .
18n both cases assumption (v) holds true when | D | and C η are sufficiently small.(vi) the same as in the previous subsection, once one notices that k ξ ′ i,l ( u )[ v ] k ≤ Z Ω (cid:0) v i |∇ η l | + η l |∇ v i | (cid:1) dx ≤ ( C η + 1) k v k . Proof of Theorem 1.2.
Since β ij = β ji , system (1.3) is variational, with poten-tial F ( u ) = k X i =1 µ i u i + X j = i β ij u i u j . It is easy to check that it satisfies assumptions (F1), (F2), (F4) with p = 4 < ∗ and δ = 2. If β ij ≤ i, j , then it also satisfies (F3), so that Theorem3.5 immediately applies. Since (F3) is used only in the estimate in Lemma 3.6(and in its counterpart in Lemma 3.1) we show how to replace that argumentin case β ij ≤ ¯ β for every i, j , with ¯ β positive and sufficiently small. We have Z Ω l ∂ i F ( u ) u i dx = Z Ω l µ i u i + X j = i β ij u i u j dx ≤ µ i Z Ω l u i dx + ¯ βC S ( B, R , where the last term is arbitrarily small when ¯ β is. References [1] A. Ambrosetti. Critical points and nonlinear variational problems.
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Dipartimento di Matematica e Applicazioni, Universit`a degli Studi di Milano-Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano, Italy [email protected]@polimi.it