Multi-bump solutions for Choquard equation with deepening potential well
aa r X i v : . [ m a t h . A P ] A p r Multi-bump solutions for Choquard equation withdeepening potential well
Claudianor O. Alves a ∗ , Alânnio B. Nóbrega a † , Minbo Yang b ‡ a. Universidade Federal de Campina GrandeUnidade Acadêmica de MatemáticaCEP: 58429-900, Campina Grande - Pb, Brazil b. Department of Mathematics, Zhejiang Normal UniversityJinhua, 321004, P. R. China.
Abstract
In this paper we study the existence of multi-bump solutions for the following Choquardequation − ∆ u + ( λa ( x ) + 1) u = (cid:0) | x | µ ∗ | u | p (cid:1) | u | p − u in R , where µ ∈ (0 , , p ∈ (2 , − µ ) , λ is a positive parameter and the nonnegative continuousfunction a ( x ) has a potential well Ω := int ( a − (0)) which possesses k disjoint boundedcomponents Ω := ∪ kj =1 Ω j . We prove that if the parameter λ is large enough, then the equationhas at least k − multi-bump solutions. Mathematics Subject Classifications (2010):
Keywords:
Choquard equation, multi-bump solution, variational methods.
The nonlinear Choquard equation − ∆ u + V ( x ) u = (cid:16) | x | µ ∗ | u | p (cid:17) | u | p − u in R , (1.1) p = 2 and µ = 1 , goes back to the description of the quantum theory of a polaron at rest by S.Pekar in 1954 [30] and the modeling of an electron trapped in its own hole in 1976 in the work ofP. Choquard, as a certain approximation to Hartree-Fock theory of one-component plasma [20]. Insome particular cases, this equation is also known as the Schrödinger-Newton equation, which wasintroduced by Penrose in his discussion on the selfgravitational collapse of a quantum mechanicalwave function [31].The existence and qualitative properties of solutions of (1.1) have been widely studied in thelast decades. In [20], Lieb proved the existence and uniqueness, up to translations, of the groundstate. Later, in [22], Lions showed the existence of a sequence of radially symmetric solutions.In [12, 25, 26] the authors showed the regularity, positivity and radial symmetry of the ground ∗ C. O. Alves was partially supported by CNPq/Brazil 301807/2013-2 and INCT-MAT, [email protected] † [email protected] ‡ M. Yang is the corresponding author, he was partially supported by NSFC (11571317, 11271331) andZJNSF(LY15A010010), [email protected] V is a continuous periodic function with inf R V ( x ) > , noticing that the nonlocal term isinvariant under translation, we can obtain easily the existence result by applying the Mountain PassTheorem, see [3] for example. For periodic potential V that changes sign and lies in the gap of thespectrum of the Schrödinger operator − ∆+ V , the problem is strongly indefinite, and the existenceof solution for p = 2 was considered in [8] by reduction arguments. For a general case, Ackermann[3] proposed a new approach to prove the existence of infinitely many geometrically distinct weaksolutions. For other related results, we refer the readers to [11, 18] for the existence of sign-changingsolutions, [28, 34, 37] for the existence and concentration behavior of the semiclassical solutionsand [29] for the critical nonlocal part with respect to the Hardy-Littlewood-Sobolev inequality.In the present paper, we are interested in the nonlinear Choquard equation with deepeningpotential well − ∆ u + ( λa ( x ) + 1) u = (cid:16) | x | µ ∗ | u | p (cid:17) | u | p − u in R , ( C ) λ where µ ∈ (0 , , p ∈ (2 , − µ ) and a ( x ) is a nonnegative continuous function with Ω = int ( a − (0)) being a non-empty bounded open set with smooth boundary ∂ Ω . Moreover, Ω has k connectedcomponents, more precisely, Ω = k [ j =1 Ω j (1.2)with dist (Ω i , Ω j ) > for i = j. (1.3)Moreover, we suppose that there exists M > such that |{ x ∈ R ; a ( x ) ≤ M }| < + ∞ . (1.4)Hereafter, if A ⊂ R is a mensurable set, | A | denotes its Lebesgue’s measure. The purpose of thepresent paper is to study the existence and the asymptotic shape of the solutions for ( C ) λ when λ is large enough, more precisely, we will show the existence of multi-bump type solutions.The motivation of the present paper arises from the results for the local Schrödinger equationswith deepening potential well − ∆ u + ( λa ( x ) + b ( x )) u = | u | p − u in R N , (1.5)where a ( x ) , b ( x ) are suitable continuous functions and p ∈ (2 , NN − if N ≥ ; p ∈ (1 , ∞ ) if N = 1 , . In [5], for b ( x ) = 1 , Bartsch and Wang proved the existence of a least energy solutionfor large λ and that the sequence of solutions converges strongly to a least energy solution for aproblem in bounded domain. They also showed the existence of at least cat Ω positive solutionsfor large λ , where Ω = int ( a − (0)) , and the exponent p is close to the critical exponent. The sameresults were also established by Clapp and Ding [10] for critical growth case. We also refer to [6] fornonconstant b ( x ) > , where the authors prove the existence of k solutions that may change signfor any k and λ large enough. For other results related to Schrödinger equations with deepeningpotential well, we may refer the readers to [33, 32, 36]The existence and characterization of the solutions for problem (1.5) with large parameter λ were considered in [1, 15], by supposing that a ( x ) has a potential well Ω = int ( a − (0)) consisting of k disjoint bounded components Ω , · · · , Ω k , the authors studied the multiplicity and multi-bumpshape of the solutions associated to the number of the components of the domain Ω = int ( a − (0)) .In [15], by using of penalization ideas developed in [14], Ding and Tanaka were able to overcome2he loss of compactness and then they applied the deformation flow arguments found in [9, 35]to prove the existence at least k − solutions u λ for large values of λ . More precisely, for eachnon-empty subset Γ of { , . . . , k } , it was proved that, for any sequence λ n → ∞ one can extract asubsequence ( λ n i ) such that ( u λ ni ) converges strongly in H (cid:0) R N (cid:1) to a function u , which satisfies u = 0 outside Ω Γ = [ j ∈ Γ Ω j and u | Ω j , j ∈ Γ , is a least energy solution for − ∆ u + u = | u | p − u, in Ω j ,u ∈ H (cid:0) Ω j (cid:1) , u > , in Ω j . (1.6)As we all know, the problem (1.6) on bounded domain plays an important role in the study ofmulti-bump shaped solutions for problem (1.5). By using of "gluing" techniques, Ding and Tanakaused the ground states of problem (1.6) as building bricks to construct minimax values and thenproved the existence of multibump solutions by deformation flow arguments.From the commentaries above, it is quite natural to ask if the results in [1, 15] still hold forthe generalized Choquard equation. Unfortunately, we can not draw a similar conclusion in astraight way, since the nonlinearity of the generalized Choquard equation is a nonlocal one. For Γ = { , · · · , l } with l ≤ k and Ω Γ = [ j ∈ Γ Ω j , it is easy to see that Z Ω Γ (cid:16) Z Ω Γ | u | p | x − y | µ dy (cid:17) | u | p dx = l X i =1 Z Ω i (cid:16) Z Ω i | u | p | x − y | µ dy (cid:17) | u | p dx. Thus, it is impossible to repeat the same arguments explored in [15] to use the least energy solutionof − ∆ u + u = (cid:16) Z Ω j | u | p | x − y | µ dy (cid:17) | u | p − u, in Ω j ,u = 0 , in Ω j ,u ∈ H (cid:0) Ω j (cid:1) , j ∈ Γ , ( C ) j for building the multi-bump solutions. For the generalized Choquard equation, it can be observedthat the equation − ∆ u + u = (cid:16) Z Ω Γ | u | p | x − y | µ dy (cid:17) | u | p − u, in Ω Γ ,u ∈ H (cid:0) Ω Γ (cid:1) , ( C ) ∞ , Γ plays the role of the limit problem for equation ( C ) λ as λ goes to infinity. Moreover, noticing thatthe solution may disappear on some component, for the Dirichlet problem of Choquard equationwith components, it is not easy to prove the existence of the least energy solution that is nonzeroon each component Ω j , j ∈ Γ . In order to find this type of least energy solution we will study theminimizing problem on a subset of the Nehari manifold, see Section 2 for more details.Here, we will also avoid the penalization arguments found in [14], because by using this methodwe are led to assume more restrictions on the constants µ and p. For that reason, instead of thepenalization method, we will follow the approach explored by Alves and Nóbrega in [2], whichshowed the existence of multi-bump solution for problem (1.5) driven by the biharmonic operator.Thus, as in [2], we will work directly with the energy functional associated with ( C ) λ , and we willmodify in a different way the set of pathes where Deformation Lemma is used, see Sections 5 and6 for more details.To prove the existence of positive multi-bump solutions for ( C ) λ , the first step is to considerthe limit Dirichlet problem ( C ) ∞ , Γ and to look for the existence of least energy solution that isnonzero on each component Ω j , j ∈ Γ . Having this in mind, we proved the following result.3 heorem 1.1. Suppose that µ ∈ (0 , and p ∈ [2 , − µ ) . Then problem ( C ) ∞ , Γ possesses a leastenergy solution u that is nonzero on each component Ω j of Ω Γ , j ∈ Γ . Using the above theorem, we are able to state our main result.
Theorem 1.2.
Suppose that µ ∈ (0 , and p ∈ (2 , − µ ) . There exists a constant λ > , such thatfor any non-empty subset Γ ⊂ { , · · · , k } and λ ≥ λ , the problem (cid:0) C (cid:1) λ has a positive solution u λ ,which possesses the following property: For any sequence λ n → ∞ we can extract a subsequence ( λ n i ) such that ( u λ ni ) converges strongly in H ( R ) to a function u , which satisfies u = 0 outside Ω Γ = [ j ∈ Γ Ω j , and u | ΩΓ is a least energy solution for ( C ) ∞ , Γ in the sense of Theorem . Remark 1.3.
Noting that ( u λ ni ) converges strongly in H ( R ) to a function u , which is zerooutside Ω Γ and nonzero on each component Ω j , j ∈ Γ . In this way, we can conclude that thesolutions ( u λ ) have the shape of multi-bump if λ is large enough. Remark 1.4.
By the Hardy-Littlewood-Sobolev inequality, the natural interval for considering theChoquard equation is ( 6 − µ , − µ ) , however, the case − µ < p ≤ is not considered in Theorem . This is due to the fact that the method applied here do have some limitations in proving theintersection property for the pathes and the set M Γ defined in Section 5. Inspired by a recent paperby Ghimenti, Moroz and Van Schaftingen [19], we will consider the case p = 2 in a future paperby approximation with p ↓ . In order to apply variational methods to obtain the solutions for problems ( C ) λ and ( C ) ∞ , Γ ,the following classical Hardy-Littlewood-Sobolev inequality will be frequently used. Proposition 1.5. [21] [ Hardy − Littlewood − Sobolev inequality ] :Let s, r > and < µ < with /s + µ/ /r = 2 . Let f ∈ L s ( R ) and h ∈ L r ( R ) . Thereexists a sharp constant C ( s, µ, r ) , independent of f, h , such that Z R Z R f ( x ) h ( y ) | x − y | µ dydx ≤ C ( s, µ, r ) | f | s | h | r . In the sequel, we fix E λ = (cid:0) E, k · k λ (cid:1) where E = (cid:26) u ∈ H ( R ) ; Z R a ( x ) | u | dx < ∞ (cid:27) , and k u k λ = (cid:18)Z R ( |∇ u | + ( λa ( x ) + 1) | u | ) dx (cid:19) . Obviously, E λ is a Hilbert space, E λ ֒ → H ( R ) continuously for λ ≥ and E λ is compactlyembedded in L sloc ( R ) , for all ≤ s < . We will study the existence of solutions for problem ( C ) λ by looking for critical points of the energy functional I λ : E λ → R given by I λ ( u ) = 12 Z R (cid:0) |∇ u | + (cid:0) λa ( x ) + 1 (cid:1) u (cid:1) dx − p Z R (cid:16) | x | µ ∗ | u | p (cid:17) | u | p dx. For p ∈ ( 6 − µ , − µ ) , the Hardy-Littlewood-Sobolev inequality and the Sobolev embeddings implythat the functional I λ ∈ C ( E λ , R ) with I ′ λ ( u ) v = Z R ∇ u ∇ v + ( λa ( x ) + 1) uvdx − Z R (cid:16) | x | µ ∗ | u | p (cid:17) | u | p − uvdx, ∀ u, v ∈ E λ . I λ are in fact the weak solutions for problem ( C ) λ .This paper is organized as follows. In Section 2, we study a nonlocal problem set on boundeddomain with 2 disjoint components for simplicity. By minimizing and deformation flow arguments,we are able to prove the existence of least energy solution which is nonzero on each component.In Section 3, we adapt the method used in [2] for the nonlocal situation, which permits us toprove that the energy functional satisfies the ( P S ) condition for λ large enough. In Section 4, westudy the behavior of ( P S ) ∞ sequence. In Section 5 and 6, we adapt the deformation flow methodto establish the existence of a special critical point, which is crucial for showing the existence ofmulti-bump solutions for λ large enough. ( C ) ∞ , Γ First, we need to study the Dirichlet problem ( C ) ∞ , Γ with several components and investigate theexistence of least energy solution that is nonzero on each component. The main idea is to provethat the energy functional associated to ( C ) ∞ , Γ defined by I Γ ( u ) = 12 Z Ω Γ ( |∇ u | + | u | ) dx − p Z Ω Γ (cid:16) Z Ω Γ | u | p | x − y | µ dy (cid:17) | u | p dx (2.1)achieves a minimum value on M Γ = { u ∈ N Γ : I ′ Γ ( u ) u i = 0 and u i = 0 , i ∈ Γ } where Γ ⊂ { , · · · , k } , u i = u | Ω i and N Γ is the Nehari manifold of I Γ defined by N Γ = { u ∈ H (Ω Γ ) \ { } : I ′ Γ ( u ) u = 0 } . More precisely, we will prove that there is w ∈ M Γ such that I Γ ( w ) = inf u ∈M Γ I Γ ( u ) and I ′ Γ ( w ) = 0 . (2.2)Hereafter, we say that w ∈ H (Ω Γ ) , satisfying w i = w | Ω i = 0 , i ∈ Γ , is a least energy solution for ( C ) ∞ , Γ if the above condition (2.2) holds. This feature will be used to characterize the multi-bumpshape of the solutions of ( C ) λ . Without loss of generality, we will only consider Γ = { , } forsimplicity. Moreover, we denote by Ω , M and N the sets Ω Γ , M Γ and N Γ respectively, and I Γ will be denoted by I . Thereby, Ω = Ω ∪ Ω , M = { u ∈ N : I ′ ( u ) u i = 0 and u i = 0 , i = 1 , . } and N = { u ∈ H (Ω) \ { } : I ′ ( u ) u = 0 } . In what follows, we denote by || || , || || and || || the norms in H (Ω) , H (Ω ) and H (Ω ) givenby || u || = (cid:18)Z Ω ( |∇ u | + | u | ) dx (cid:19) , || u || = (cid:18)Z Ω ( |∇ u | + | u | ) dx (cid:19) and || u || = (cid:18)Z Ω ( |∇ u | + | u | ) dx (cid:19) respectively.The following Lemma shows that the set M is not empty.5 emma 2.1. Let < µ < , ≤ p < − µ and v ∈ H (Ω) with v j = 0 for j = 1 , , thenthere exists ( β , β ) ∈ (0 , + ∞ ) such that β v + β v ∈ M which means M 6 = ∅ and moreover, c = inf u ∈M I ( u ) > .Proof. Fix v ∈ H (Ω) with v i = 0 for i = 1 , , and for the case p = 2 , without loss of generality,we may additionally assume that k v i k i = Z Ω i (cid:16) Z Ω i | v i | | x − y | µ dy (cid:17) | v i | dx for i = 1 , . (2.3)Adapt some ideas in [18] and [19] by changing variables t j = s p j , we define the function G ( s , s ) = I ( s p v + s p v ) = s p k v k + s p k v k − p Z Ω (cid:18) | x | µ ∗ ( s | v | p + s | v | p ) (cid:19) dx. As G is a continuous function and G ( s , s ) → as | ( s , s ) | → + ∞ , we have that G possesses aglobal maximum point ( a, b ) ∈ [0 , + ∞ ) . However, as G is strictly concave function, it follows that ( a, b ) ∈ (0 , + ∞ ) , ( a, b ) is the unique global maximum point and ∇ G ( a, b ) = (0 , , which impliesthat M 6 = ∅ . Here, we would like to point out that if p > , it is easy to check that a, b = 0 . Whilefor the case p = 2 , we are able to show this fact only with the restriction (2.3). In fact, argue bycontradiction that a = 0 , notice that (0 , b ) is the maximum point of G then there holds k v k = b Z Ω (cid:16) Z Ω | v | | x − y | µ dy (cid:17) | v | dx, therefore b = 1 . Consider the function g : [0 , + ∞ ) → R given by g ( t ) = G (0 , b + αt ) , where α is to be determined later. A direct computation shows that g ′ (0) = 12 k v k − b Z Ω Z Ω | v | ( y ) | v | ( x ) | x − y | µ dydx + α (cid:16) k v k − b Z Ω (cid:16) Z Ω | v | | x − y | µ dy (cid:17) | v | dx (cid:17) . Consequently, if α is suitably chosen, we have g ′ (0) > , which obviously is a contradiction.Next, we will show that c > . To begin with, we recall that if w ∈ M , then k w k = Z Ω Z Ω | w ( y ) | p | w ( x ) | p | x − y | µ dydx. Using the Hardy-Littlewood-Sobolev inequality, there is
C > such that k w k ≤ C k w k . As k w k 6 = 0 , the last inequality yields there is τ > satisfying k w k ≥ τ, ∀ w ∈ M . From this, I ( w ) = I ( w ) − p I ′ ( w ) w = (cid:18) − p (cid:19) k w k ≥ (cid:18) − p (cid:19) τ > , ∀ w ∈ M . and so, c ≥ (cid:18) − p (cid:19) τ > . 6et us state more a technical lemma. Lemma 2.2.
Let < µ < , ≤ p < − µ and ( w n ) be a bounded sequence in M with w n ⇀ w in H (Ω) . If k w n,j k 6→ , then w j = 0 , where w n,j = w n | Ω j and w j = w | Ω j for j = 1 , .Proof. Assume by contradiction that w = 0 . By the Hardy-Littlewood-Sobolev inequality andthe Sobolev embeddings, we see that Z Ω j (cid:18)Z Ω | w n | p | x − y | µ dy (cid:19) | w n, | p dx → . On the other hand, as I ′ ( w n )( w n,j ) = 0 , or equivalently k w n, k = Z Ω (cid:18)Z Ω | w n | p | x − y | µ dy (cid:19) | w n, | p dx, we derive that k w n, k → which is an absurd. The case w = 0 is made of similar way.Now, we are able to show the existence of least energy solution for ( C ) ∞ , Γ . From Lemma 2.1, c > and there is a sequence ( w n ) ⊂ M such that lim n I ( w n ) = c . It is easy to see that ( w n ) is a bounded sequence. Hence, without loss of generality, we maysuppose that w n ⇀ w in H (Ω) and w n → w in L q (Ω) ∀ q ∈ [1 , , as n → ∞ . By consideringthe function G ( s , s ) = I ( s p ( w n ) + s p ( w n ) ) = s p k ( w n ) k + s p k ( w n ) k − p Z Ω (cid:18) | x | µ ∗ ( s | ( w n ) | p + s | ( w n ) | p ) (cid:19) dx. we know by the previous study that ∇ G (1 ,
1) = (0 , . As G is strictly concave function, (1 , isits global maximum point. Thus, I ( w n ) = I (( w n ) + ( w n ) ) = max t,s ≥ I ( t ( w n ) + s ( w n ) ) . Using the above information, we also know that w j = 0 for j = 1 , . Then, by Lemma 2.1 thereare t , t > such that t w + t w ∈ M , and so, c ≤ I ( t w + t w ) . By using the fact that w n ⇀ w in H (Ω) and the compact Sobolev embeddings, we get I ( t w + t w ) ≤ lim inf n → + ∞ I ( t ( w n ) + t ( w n ) ) ≤ lim inf n → + ∞ I ( w n ) = c , from where it follows that c = I ( t w + t w ) with t w + t w ∈ M . w ∗ = t w + t w is a critical point for I . Assume by contradictionthat k I ′ ( w ∗ ) k > and fix α > such that k I ′ ( w ∗ ) k ≥ α. Moreover, we will fix r > small enough such that if ( t, s ) ∈ B = B r (1 , ⊂ R , then there existssome ǫ > such that I ( t p ( w ∗ ) + s p ( w ∗ ) ) < c − ǫ , ∀ ( t, s ) ∈ ∂B. (2.4)In the sequel we fix ǫ ∈ (0 , ǫ ) and δ > small emough such that k I ′ ( u ) k ≥ α ≥ ǫδ ∀ u ∈ I − { [ c − ǫ, c + 2 ǫ ] } ∩ S where S = { t p ( w ∗ ) + s p ( w ∗ ) : ( t, s ) ∈ B } . By using the Deformation Lemma, there exists a continuous map η : H (Ω) → H (Ω) , such that η ( u ) = u ∀ u / ∈ I − { [ c − ǫ, c + 2 ǫ ] } ∩ S (2.5)and η ( I c + ǫ ∩ S ) ⊂ I c − ǫ ∩ S δ (2.6)where S δ = { v ∈ H (Ω) : dist ( v, S ) ≤ δ } . In the sequel, we fix δ > of a way that v ∈ S δ ⇒ v , v = 0 . (2.7)Now, setting γ ( t, s ) = η ( t p ( w ∗ ) + s p ( w ∗ ) ) , (2.4) and (2.5) imply that γ ( s, t ) = t p ( w ∗ ) + s p ( w ∗ ) , ∀ ( s, t ) ∈ ∂B. (2.8)Moreover, since max t,s ≥ I ( t p ( w ∗ ) + s p ( w ∗ ) ) = I ( t () w ∗ ) + t ( w ∗ ) ) = c , by (2.6), we know I ( γ ( t, s )) ≤ c − ǫ. Claim 2.3.
There is ( t , s ) ∈ B such that (cid:0) I ′ ( γ ( t , s ))( γ ( t , s ) ) , I ′ ( γ ( t , s ))( γ ( t , s ) ) (cid:1) = (0 , . Assuming for a moment the claim is true, we deduce that γ ( t , s ) ∈ M , and so, c ≤ I ( γ ( t , s )) ≤ c − ǫ which is absurd. Here, (2.7) was used to ensure that γ ( t , s ) j = 0 for j = 1 , . Consequently, w ∗ = t w + t w is a critical point for I . Proof of Claim 2.3:
First of all, note that (cid:0) I ′ ( γ ( t, s ))( γ ( t, s ) ) , I ′ ( γ ( t, s ))( γ ( t, s ) ) (cid:1) = (0 , ⇔ (cid:0) t I ′ ( γ ( t, s ))( γ ( t, s ) ) , s I ′ ( γ ( t, s ))( γ ( t, s ) ) (cid:1) = (0 , (cid:0) t I ′ ( γ ( t, s ))( γ ( t, s ) ) , s I ′ ( γ ( t, s ))( γ ( t, s ) ) (cid:1) = (cid:0) t I ′ ( γ ( t, s ))( γ ( t, s ) ) , s I ′ ( γ ( t, s ))( γ ( t, s ) ) (cid:1) ∀ ( t, s ) ∈ ∂B where γ ( t, s ) = t p ( w ∗ ) + s p ( w ∗ ) ∀ ( s, t ) ∈ B. Considering the function G ( t, s ) = I ( t p ( w ∗ ) + s p ( w ∗ ) ) , we deduce that ( 1 t I ′ ( γ ( t, s ))( γ ( t, s ) ) , s I ′ ( γ ( t, s ))( γ ( t, s ) )) = ∇ G ( t, s ) ∀ ( t, s ) ∈ B. Since G is is strictly concave function and ∇ G (1 ,
1) = (0 , , it follows that > h∇ G ( t, s ) − ∇ G (1 , , ( t, s ) − (1 , i = h∇ G ( t, s ) , ( t, s ) − (1 , i ∀ ( s, t ) = (1 , , and so, > h∇ G ( t, s ) , ( t, s ) − (1 , i for | ( s, t ) − (1 , | = r. Setting H : R → R by H ( t, s ) = (cid:0) t I ′ ( γ ( t, s ))( γ ( t, s ) ) , s I ′ ( γ ( t, s ))( γ ( t, s ) ) (cid:1) and f ( t, s ) = H ( t + 1 , s + 1) , we have that > h f ( t, s ) , ( t, s ) i for | ( s, t ) | = r. By using the Brouwer’s fixed point Theorem, we know there exists ( t ∗ , s ∗ ) ∈ B r (0 , such that f ( t ∗ , s ∗ ) = (0 , , that is, H ( t ∗ + 1 , s ∗ + 1) = (0 , , from where it follows that there is ( t , s ) ∈ B such that (cid:0) I ′ ( γ ( t, s ))( γ ( t, s ) ) , I ′ ( γ ( t, s ))( γ ( t, s ) ) (cid:1) = (0 , , which completes the proof of the claim. ( P S ) c condition for I λ In this section, we will prove some convergence properties for the ( P S ) sequences of the functional I λ . Our main goal is to prove that, for given c ≥ independent of λ , the functional I λ satisfies the ( P S ) d condition for d ∈ [0 , c ) , provided that λ is large enough. Lemma 3.1.
Let ( u n ) ⊂ E λ be a ( P S ) c sequence for I λ , then ( u n ) is bounded. Furthermore, c ≥ . Proof.
Since ( u n ) is a ( P S ) c sequence, I λ ( u n ) → c and I ′ λ ( u n ) → . Then, for n large enough I λ ( u n ) − p I ′ λ ( u n ) u n ≤ c + 1 + k u n k λ . (3.1)9n the other hand, I λ ( u n ) − p I ′ λ ( u n ) u n = (cid:18) − p (cid:19) k u n k λ . (3.2)Therefore, from ( ) and ( ) we get the inequality below (cid:18) − p (cid:19) k u n k λ ≤ c + 1 + k u n k λ , which shows the boundedness of ( u n ) . Thereby, by (3.2), ≤ (cid:18) − p (cid:19) k u n k λ ≤ c + o n (1) , (3.3)and the lemma follows by taking the limit of n → + ∞ . Corollary 3.2.
Let ( u n ) ⊂ E λ be a ( P S ) sequence for I λ . Then u n → in E λ .Proof. An immediate consequence of the arguments used in the proof of Lemma 3.1.Next we prove a splitting property for the functional I λ , which is related to the Brezis-Liebtype Lemma for nonlocal nonlinearities [3, 27]. Lemma 3.3.
Let c ≥ and ( u n ) be a ( P S ) c sequence for I λ . If u n ⇀ u in E λ , then I λ ( v n ) − I λ ( u n ) + I λ ( u ) = o n (1) (3.4) I ′ λ ( v n ) − I ′ λ ( u n ) + I ′ λ ( u ) = o n (1) , (3.5) where v n = u n − u. Furthermore, ( v n ) is a ( P S ) c − I λ ( u ) sequence.Proof. First of all, note that I λ ( v n ) − I λ ( u n ) + I λ ( u ) = 12 (cid:0) k v n k λ − k u n k λ + k u k λ (cid:1) −− p Z R (cid:18)Z R | v n ( y ) | p | v n ( x ) | p − | u n ( y ) | p | u n ( x ) | p + | u ( y ) | p | u ( x ) | p | x − y | µ dy (cid:19) dx. Since u n ⇀ u in E λ , we have I λ ( v n ) − I λ ( u n ) + I λ ( u ) = o n (1) + 12 p Z R (cid:18)Z R | v n ( y ) | p ( − | v n ( x ) | p + | u n ( x ) | p − | u ( x ) | p ) | x − y | µ dy (cid:19) dx + 12 p Z R (cid:18)Z R | u n ( y ) | p ( − | v n ( x ) | p + | u n ( x ) | p − | u ( x ) | p ) | x − y | µ dy (cid:19) dx (3.6) + 12 p Z R (cid:18) Z R | u ( y ) | p ( − | v n ( x ) | p + | u n ( x ) | p − | u ( x ) | p ) | x − y | µ dy (cid:19) dx + 1 p Z R (cid:18)Z R | v n ( y ) | p | u ( x ) | p | x − y | µ dy (cid:19) dx. By the Hardy-Litllewood-Sobolev inequality, Z R Z R | v n ( y ) | p (cid:16) | v n ( x ) | p − | u n ( x ) | p + | u ( x ) | p (cid:17) | x − y | µ dy dx ≤ C | v n | p p − µ (cid:18)Z R (cid:12)(cid:12) | v n ( x ) | p − | u n ( x ) | p + | u ( x ) | p (cid:12)(cid:12) − µ dx (cid:19) − µ . Notice that Z R (cid:12)(cid:12)(cid:12) | v n ( x ) | p − | u n ( x ) | p + | u ( x ) | p (cid:12)(cid:12)(cid:12) − µ dx = Z B R (0) (cid:12)(cid:12)(cid:12) | v n ( x ) | p − | u n ( x ) | p + | u ( x ) | p (cid:12)(cid:12)(cid:12) − µ dx + (3.7) Z R \ B R (0) (cid:12)(cid:12)(cid:12) | v n ( x ) | p − | u n ( x ) | p + | u ( x ) | p (cid:12)(cid:12)(cid:12) − µ dx. where, R > will be fixed subsequently. As u n ⇀ u in E λ , we know that10 u n → u, in L p − µ ( B R (0)); • u n ( x ) → u ( x ) a.e. in R , and there is h ∈ L p − µ ( B R (0)) such that | u n ( x ) | ≤ h ( x ) a.e. in R . From this, | v n ( x ) | p − | u n ( x ) | p + | u ( x ) | p → a.e. in R , and (cid:12)(cid:12)(cid:12) | v n ( x ) | p − | u n ( x ) | p + | u ( x ) | p (cid:12)(cid:12)(cid:12) − µ ≤ (2 p + 1) − µ ( h ( x ) + | u ( x ) | ) p − µ ∈ L ( B R (0)) . Thus, by the Lebesgue Dominated Convergence Theorem, Z B R (0) (cid:12)(cid:12)(cid:12) | v n ( x ) | p − | u n ( x ) | p + | u ( x ) | p (cid:12)(cid:12)(cid:12) − µ dx → . (3.8)Furthermore, we also have (cid:12)(cid:12)(cid:12) | u n ( x ) − u ( x ) | p − | u n ( x ) | p (cid:12)(cid:12)(cid:12) ≤ p p − ( | u n ( x ) | p − | u ( x ) | + | u ( x ) | p ) , and so, Z R \ B R (0) (cid:12)(cid:12)(cid:12) | u n ( x ) − u ( x ) | p −| u n ( x ) | p (cid:12)(cid:12)(cid:12) − µ dx ≤ C Z R \ B R (0) | u n ( x ) | p − − µ | u ( x ) | − µ dx + C Z R \ B R (0) | u ( x ) | p − µ dx. For ε > , we can choose R > such that Z R \ B R (0) | u ( x ) | p − µ dx ≤ ε, the Hölder inequality combined with the boundedness of ( u n ) implies that Z R \ B R (0) (cid:12)(cid:12)(cid:12) | v n ( x ) | p − | u n ( x ) | p + | u ( x ) | p (cid:12)(cid:12)(cid:12) − µ dx ≤ ε. (3.9)Gathering together the boundedness of ( v n ) and (3.7)-(3.9), we deduce that Z R (cid:12)(cid:12)(cid:12) | v n ( x ) | p − | u n ( x ) | p + | u ( x ) | p (cid:12)(cid:12)(cid:12) − µ dx → . To finish the proof, we need to prove that Z R (cid:18)Z R | v n ( y ) | p | u ( x ) | p | x − y | µ dy (cid:19) dx → . Once v n ⇀ in E λ and p ∈ (2 , − µ ) , the sequence ( | v n | p ) is bounded in L − µ ( R ) . As v n ( x ) → a.e. in R , we ensure that | v n | p converges weakly to in L − µ ( R ) . Using again the Hardy-Littlewood-Sobolev inequality, we know that the linear functional F : L − µ ( R ) → R definedby F ( w ) = Z R (cid:0) | x | µ ∗ w (cid:1) | u ( x ) | p dx is continuous. Consequently, F ( | v n | p ) → , or equivalently, Z R (cid:0) | x | µ ∗ | v n ( x ) | p (cid:1) | u ( x ) | p dx → , and the proof is complete. 11 emma 3.4. Let ( u n ) be a ( P S ) c sequence for I λ . Then c = 0 , or there exists c ∗ > , independentof λ, such that c ≥ c ∗ , for all λ > . Proof.
By Lemma 3.1, we know c ≥ . Suppose that c > . On one hand, we know c + o n (1) k u n k λ = I λ ( u n ) − p I ′ λ ( u n ) u n ≥ (cid:18) p − p (cid:19) k u n k λ , equivalently, lim sup n → + ∞ k u n k λ ≤ pcp − . (3.10)On the other hand, the Hardy-Littlewood-Sobolev inequality together with the Sobolev embeddingtheorems imply that I ′ λ ( u n ) u n ≥ k u n k λ − K k u n k pλ , where K is a positive constant. Thus, there exists δ > such that I ′ λ ( u n ) u n ≥ || u n || λ , for || u n || λ < δ. (3.11)Consider c ∗ = δ p − p and c < c ∗ . Then it follows that k u n k λ ≤ δ (3.12)for n large enough. Hence, I ′ λ ( u n ) u n ≥ k u n k λ , and thus k u n k λ → . Thereby, I λ ( u n ) → I λ (0) = 0 , which contradicts the fact that ( u n ) is a ( P S ) c sequence with c > . Therefore, c ≥ c ∗ . Lemma 3.5.
Let ( u n ) be a ( P S ) c sequence for I λ . Then, there exists δ > independent of λ, such that lim inf n → + ∞ | u n | p p − µ ≥ δ c. Proof.
Note that c = lim n → + ∞ (cid:18) I λ ( u n ) − I ′ λ ( u n ) u n (cid:19) = (cid:18) − p (cid:19) lim n → + ∞ Z R (cid:16) | x | µ ∗ | u n | p (cid:17) | u n | p dx, by the Hardy-Littlewood-Sobolev inequality, we obtain c ≤ (cid:18) − p (cid:19) K lim inf n → + ∞ | u n | p p − µ . Therefore, the conclusion follows by setting δ = (cid:18) p − p (cid:19) K − > . emma 3.6. Let c > be a constant independent of λ . Given ε > , there exist Λ = Λ( ε ) and R = R ( ε, c ) such that, if ( u n ) is a ( P S ) c sequence for I λ with c ∈ [0 , c ] , then lim sup n → + ∞ | u n | p p − µ ,B cR (0) ≤ ε, ∀ λ ≥ Λ . Proof.
For
R > , consider A ( R ) = { x ∈ R / | x | > R and a ( x ) ≥ M } and B ( R ) = { x ∈ R / | x | > R and a ( x ) < M } . Then, Z A ( R ) u n dx ≤ λM + 1) Z R ( λa ( x ) + 1) u n dx ≤ λM + 1) || u n || λ (3.13) ≤ λM + 1) "(cid:18) − p (cid:19) − c + o n (1) ≤ λM + 1) "(cid:18) − p (cid:19) − c + o n (1) . Once c is independent of λ, by ( ) there is Λ > such that lim sup n → + ∞ Z A ( R ) u n dx < ε , ∀ λ ≥ Λ . (3.14)On the other hand, using the Hölder inequality for s ∈ [1 , and the continuous embedding E λ ֒ → L s ( R ) , we see that Z B ( R ) u n dx ≤ β || u n || λ | B ( R ) | s ′ ≤ c (cid:18) − p (cid:19) − | B ( R ) | s ′ + o n (1) , where β is a positive constant. Now, by assumption (1.4) on the potential a ( x ) , we know that | B ( R ) | → , when R → + ∞ , then we can choose R large enough such that lim sup n → + ∞ Z B ( R ) u n dx < ε . (3.15)Using (3.14) and (3.15), we obtain that lim sup n → + ∞ Z R u n dx < ε. The last inequality combined with interpolation implies that lim sup n → + ∞ Z R \ B R (0) | u n | p − µ dx < ε, λ > Λ , by increasing R and Λ if necessary. 13 roposition 3.7. Given c > , independent of λ , there exists Λ = Λ( c ) > such that if λ ≥ Λ ,then I λ verifies the ( P S ) c condition for all c ∈ [0 , c ] .Proof. Let ( u n ) be a ( P S ) c sequence. Lemma 3.1 implies that ( u n ) is bounded. Passing to asubsequence if necessary, u n ⇀ u, in E λ ; u n ( x ) → u ( x ) , a.e. in R ; u n → u, in L sloc ( R ) , ≤ s < . Then, I ′ λ ( u ) = 0 and I λ ( u ) ≥ . Setting v n = u n − u, Lemma 3.3 ensures that ( v n ) is a ( P S ) d sequence with d = c − I λ ( u ) . Furthermore, ≤ d = c − I λ ( u ) ≤ c ≤ c We claim that d = 0 . Otherwise, suppose that d > . By Lemma 3.4 and Lemma 3.5, we know d ≥ c ∗ and lim inf n → + ∞ | v n | p p − µ ≥ δ c ∗ > . (3.16)Applying Lemma 3.6 with ε = δ c ∗ > , there exist Λ , R > such that lim sup n → + ∞ | v n | p p − µ ,B CR (0) ≤ δ c ∗ , for λ ≥ Λ . (3.17)Combining ( ) and ( ) , we obtain lim inf n → + ∞ | v n | p p − µ ,B R (0) ≥ δ c ∗ > , which is absurd, because as v n ⇀ in E λ , the compact embedding E λ ֒ → L p − µ ( B R (0)) impliesthat lim inf n → + ∞ | v n | p p − µ ,B r (0) = 0 . Thereby, d = 0 and ( v n ) is a ( P S ) sequence. Hence, by Corollary 3.2, v n → in E λ . Thus, I λ satisfies the ( P S ) c condition for c ∈ [0 , c ] if λ is large enough. ( P S ) ∞ condition A sequence ( u n ) ⊂ H ( R ) is called a ( P S ) ∞ sequence for the family ( I λ ) λ ≥ , if there exist d ∈ [0 , c Γ ] and a sequence ( λ n ) ⊂ [1 , ∞ ) with λ n → ∞ , such that I λ n ( u n ) → d and (cid:13)(cid:13) I ′ λ n ( u n ) (cid:13)(cid:13) E ∗ λn → , as n → ∞ . Proposition 4.1.
Suppose that < µ < , ≤ p < − µ and ( u n ) ⊂ H ( R ) is a ( P S ) ∞ sequencefor ( I λ ) λ ≥ with < d ≤ c Γ . Then, up to subsequence, there exists u ∈ H ( R ) such that u n ⇀ u in H ( R ) . Furthermore,(i) u n → u in H ( R ) ;(ii) u = 0 in R \ Ω and u ∈ H (Ω) is a solution for − ∆ u + u = (cid:16) Z Ω | u | p | x − y | µ dy (cid:17) | u | p − u in Ω; iii) λ n Z R a ( x ) | u n | → ;(iv) k u n − u k λ, Ω → ;(v) k u n k λ, R \ Ω → ;(vi) I λ n ( u n ) → Z Ω ( |∇ u | + | u | ) dx − p Z Ω (cid:16) Z Ω | u | p | x − y | µ dy (cid:17) | u | p dx .Proof. By hypothesis, I λ n ( u n ) → d and k I ′ λ n ( u n ) k E ′ λn → . Then, the same arguments employed in the proof of Lemma 3.1 imply that ( k u n k λ n ) and ( u n ) arebounded in R and H ( R ) respectively. And so, up to subsequence, there exists u ∈ H ( R ) suchthat u n ⇀ u in H ( R ) and u n ( x ) → u ( x ) for a.e. x ∈ R . Now, for each m ∈ N , we define C m = (cid:26) x ∈ R ; a ( x ) ≥ m (cid:27) . Without loss of generality, we mayassume that λ n < λ n − , ∀ n ∈ N . Thus Z C m | u n | dx ≤ mλ n Z C m (cid:0) λ n a ( x ) + 1) | u n | dx ≤ Cλ n . By Fatou’s lemma, we derive that Z C m | u | dx = 0 , which implies that u = 0 in C m , and so, u = 0 in R \ Ω . From this, we are able to prove ( i ) − ( vi ) . ( i ) By a simple computation, we see that k u n − u k λ n = I ′ λ n ( u n ) u n − I ′ λ n ( u n ) u + Z R (cid:0) | x | µ ∗ | u n | p (cid:1) | u n | p − u n ( u n − u ) dx + o n (1) , then, k u n − u k λ n = Z R (cid:0) | x | µ ∗ | u n | p (cid:1) | u n | p − u n ( u n − u ) dx + o n (1) . As in the proof of Lemma 3.3, k u n − u k λ n → , which means that u n → u in H ( R ) . ( ii ) Since u ∈ H ( R ) and u = 0 in R \ Ω , we know u ∈ H (Ω) and u | Ω j ∈ H (Ω j ) , for j ∈ { , ..., k } . Moreover, taking into account that u n → u in H ( R ) and I ′ λ n ( u n ) ϕ → for ϕ ∈ C ∞ (Ω) , we get Z Ω ( ∇ u ∇ ϕ + uϕ ) dx − Z Ω (cid:16) Z Ω | u | p | x − y | µ dy (cid:17) | u | p − uϕdx = 0 , (4.1)which shows that u | Ω is a solution for the nonlocal problem − ∆ u + u = (cid:16) Z Ω | u | p | x − y | µ dy (cid:17) | u | p − u in Ω . ( iii ) In view of (i), λ n Z R a ( x ) | u n | dx = Z R λ n a ( x ) | u n − u | dx ≤ k u n − u k λ n . λ n Z R a ( x ) | u n | dx → . ( iv ) For each j ∈ { , ..., k } , | u n − u | , Ω j , |∇ u n − ∇ u | , Ω j → . ( see ( i )) . Therefore, Z Ω ( |∇ u n | − |∇ u | ) dx → and Z Ω ( | u n | − | u | ) dx → . In view of (iii), we know Z Ω λ n a ( x ) | u n | dx → , then k u n k λ n , Ω → Z Ω ( |∇ u | + | u | ) dx. ( v ) Summarizing (i) and k u n − u k λ n → , we obtain k u n k λ n , R \ Ω → . ( vi ) We can write the functional I λ n in the following way I λ n ( u n ) = 12 Z Ω ( |∇ u n | + ( λ n a ( x ) + 1) | u n | ) dx + 12 Z R \ Ω ( |∇ u n | + ( λ n a ( x ) + 1) | u n | ) dx − p Z R \ Ω (cid:0) | x | µ ∗ | u n | p (cid:1) | u n | p dx − p Z Ω (cid:0) | x | µ ∗ | u n | p ) | u n | p dx. Using ( i ) − ( v ) , we get Z Ω ( |∇ u n | + ( λ n a ( x ) + 1) | u n | ) dx → Z Ω ( |∇ u | + | u | ) dx, Z R \ Ω ( |∇ u n | + ( λ n a ( x ) + 1) | u n | ) dx → , Z Ω (cid:0) | x | µ ∗ | u n | p (cid:1) | u n | p dx → Z Ω (cid:16) Z Ω | u | p | x − y | µ dy (cid:17) | u | p dx, Z R \ Ω (cid:0) | x | µ ∗ | u n | p (cid:1) | u n | p dx → . Therefore, we can conclude that I λ n ( u n ) → Z Ω ( |∇ u | + | u | ) dx − p Z Ω (cid:16) Z Ω | u | p | x − y | µ dy (cid:17) | u | p dx. Further propositions for c Γ In the sequel, without loss of generality, we consider
Γ = { , · · · , l } , with l ≤ k . Moreover, let usdenote by Ω ′ Γ = ∪ j ∈ Γ Ω ′ j , where Ω ′ j is an open neighborhood of Ω j with Ω ′ j ∩ Ω ′ i = ∅ if j = i . Usingthis notion, we introduce the functional I λ, Γ ( u ) = 12 Z Ω ′ Γ ( |∇ u | + ( λa ( x ) + 1) | u | ) dx − p Z Ω ′ Γ (cid:16) Z Ω ′ Γ | u | p | x − y | µ dy (cid:17) | u | p dx, which is the energy functional associated to the Choquard equation with Neumann boundarycondition − ∆ u + ( λa ( x ) + 1) u = (cid:16) Z Ω ′ Γ | u | p | x − y | µ dy (cid:17) | u | p − u, in Ω ′ Γ ,∂u∂η = 0 , on ∂ Ω ′ Γ . ( CN λ ) In what follows, we denote by c Γ the number given by c Γ = inf u ∈M Γ I Γ ( u ) where M Γ = { u ∈ N Γ : I ′ Γ ( u ) u j = 0 and u j = 0 , ∀ j ∈ Γ } with u j = u | Ω j and N Γ = { u ∈ H (Ω Γ ) \ { } : I ′ Γ ( u ) u = 0 } . Similarly, we denote by c λ, Γ the number given by c λ, Γ = inf u ∈M ′ Γ I λ, Γ ( u ) where M ′ Γ = { u ∈ N ′ Γ : I ′ λ, Γ ( u ) u j = 0 and u j = 0 , ∀ j ∈ Γ } with u j = u | Ω ′ j and N ′ Γ = { u ∈ H (Ω ′ Γ ) \ { } : I ′ λ, Γ ( u ) u = 0 } . Repeating the same arguments in Section 2, we know that there exist w Γ ∈ H (Ω Γ ) and w λ, Γ ∈ H (Ω ′ Γ ) such that I Γ ( w Γ ) = c Γ and I ′ Γ ( w Γ ) = 0 and I λ, Γ ( w λ, Γ ) = c λ, Γ and I ′ λ, Γ ( w λ, Γ ) = 0 . We have the following proposition, which describes an important relation between c Γ and c λ, Γ . Lemma 5.1.
There holds that(i) < c λ, Γ ≤ c Γ , ∀ λ ≥ ;(ii) c λ, Γ → c Γ , as λ → ∞ .Proof. ( i ) Since H (Ω Γ ) ⊂ H (Ω ′ Γ ) , it is easy to see that < c λ, Γ ≤ c Γ . ( ii ) Let λ n → ∞ . From the above commentaries, for each λ n there exists w n ∈ H (Ω ′ ) with I λ n , Γ ( w n ) = c λ n , Γ and I ′ λ n , Γ ( w n ) = 0 . (cid:0) c λ n , Γ (cid:1) is bounded, there exists ( w n i ) , subsequence of ( w n ) , such that ( I λ ni , Γ ( w n i )) convergesand I ′ λ ni , Γ ( w n i ) = 0 . Repeating the same ideas explored in the proof of Proposition 4.1, we knowthat there exists w ∈ H (Ω Γ ) \ { } ⊂ H (Ω ′ Γ ) such that w j = w | Ω j = 0 , j ∈ Γ and w n i → w in H (Ω ′ Γ ) , as n i → ∞ . Furthermore, we also have that c λ ni , Γ = I λ ni , Γ ( w n i ) → I Γ ( w ) and I ′ λ ni , Ω ′ ( w n i ) → I ′ Γ ( w ) . By the definition of c Γ , lim i c λ ni , Γ ≥ c Γ . Then, combining the last limit with conclusion (i), we can guarantee that c λ ni , Γ → c Γ , as n i → ∞ . This establishes the asserted result.In the sequel, we denote by w ∈ H (Ω Γ ) the least energy solution obtained in Section 2, thatis, w ∈ M Γ , I Γ ( w ) = c Γ and I ′ Γ ( w ) = 0 . (5.1)Changing variables by t j = s p j , it is obvious that I Γ (cid:0) t w + · · · + t l w l (cid:1) = l X j =1 t j || w j || j − p Z Ω Γ Z Ω Γ (cid:12)(cid:12) P lj =1 t j w j (cid:12)(cid:12) p | x − y | µ dy ! (cid:12)(cid:12)(cid:12) l X j =1 t j w j (cid:12)(cid:12)(cid:12) p dx = l X j =1 s p j || w j || j − p Z Ω Γ Z Ω Γ P lj =1 s j | w j | p | x − y | µ dy ! (cid:16) l X j =1 s j | w j | p (cid:17) dx. Arguing as in [18], Z Ω Γ Z Ω Γ P lj =1 s j | w j | p | x − y | µ dy ! (cid:16) l X j =1 s j | w j | p (cid:17) dx = Z Ω Γ | x | µ/ ∗ (cid:16) l X j =1 s j | w j | p (cid:17) dx. As s s /p is concave and s s is strictly convex, we concluded that the function G ( s , s , · · · , s l ) = I Γ ( s p w + · · · + s p l w l ) is strictly concave with ∇ G (1 , · · · ,
1) = 0 . Hence, (1 , · · · , is the unique global maximum pointof G on [0 , + ∞ ) l with G (1 , · · · ,
1) = c Γ . In the sequel, we denote by w ∈ H (Ω Γ ) the least energysolution obtained in Section 2, that is, w ∈ M Γ , I Γ ( w ) = c Γ and I ′ Γ ( w ) = 0 . p > , there are r > small enough and R > large enough such that I ′ Γ ( l X j =1 ,j = i t j w j ( x ) + Rw i )( Rw i ) < , for i ∈ Γ , ∀ t j ∈ [ r, R ] and j = i, (5.2) I ′ Γ ( l X j =1 ,j = i t j w j ( x ) + rw i )( rw i ) > , for i ∈ Γ , ∀ t j ∈ [ r, R ] and j = i. (5.3)and I Γ (cid:0) l X j =1 t j w j ( x ) (cid:1) < c Γ , ∀ ( t , · · · , t l ) ∈ ∂ [ r, R ] l , (5.4)where w j := w | Ω j , j ∈ Γ . Using these information, we can define γ ( t , · · · , t l )( x ) = l X j =1 t j w j ( x ) ∈ H (Ω Γ ) , ∀ ( t , · · · , t l ) ∈ [ r, R ] l . and denote by Γ ∗ the class of continuous pathes γ ∈ C (cid:0) [ r, R ] l , E λ \{ } (cid:1) which satisfies the followingconditions: ( a ) γ = γ on ∂ [ r, R ] l , and ( b ) Φ Γ ( γ ) = 12 Z R \ Ω ′ Γ (cid:0) |∇ γ | + ( λa ( x ) + 1) | γ | (cid:1) dx − p Z R \ Ω ′ Γ (cid:18) | x | µ ∗ | γ | p (cid:19) | γ | p dx ≥ , where R > > r > are the positive constants obtained in (5.2) and (5.3). Since γ ∈ Γ ∗ , weknow that Γ ∗ = ∅ . And by (a) for the path γ and (5.4), we have I λ (cid:0) γ ( t , · · · , t l ) (cid:1) < c Γ , ∀ ( t , · · · , t l ) ∈ ∂ [ r, R ] l , ∀ γ ∈ Γ ∗ . (5.5)The following lemma will be used to describe the intersection property of the paths and the set M Γ in the final section. Lemma 5.2.
For all γ ∈ Γ ∗ , there exists ( t , . . . , t l ) ∈ ( r, R ) l such that I ′ λ, Γ ( γ ( t , . . . , t l )) γ j ( t , . . . , t l ) = 0 , where γ j ( t , . . . , t l ) = γ ( t , . . . , t l ) | Ω ′ j , j ∈ Γ .Proof. Since p > and γ = γ on ∂ [ r, R ] l , by using of (5.2) and (5.3), we see that the result followsby Miranda’s Theorem [24]. In this section, we are ready to find nonnegative solutions u λ for large values of λ , which convergesto a least energy solution of ( C ) ∞ , Γ as λ → ∞ . To this end, we will prove two propositions which,together with Propositions 4.1, will help us to show the main result in Theorem 1.2.Henceforth, we denote by Θ = n u ∈ E λ : k u k λ, Ω ′ j > rτ j = 1 , · · · , l o , r was fixed in (5.2) and τ is the positive constant such that k u j k j > τ, ∀ u ∈ Υ Γ = { u ∈ M Γ : I Γ ( u ) = c Γ } and ∀ j ∈ Γ . Furthermore, I c Γ λ denotes the set I c Γ λ = (cid:8) u ∈ E λ ; I λ ( u ) ≤ c Γ (cid:9) . Fixing δ = rτ , for ξ > small enough, we set A λξ = n u ∈ Θ δ : Φ Γ ( u ) ≥ , || u || λ, R \ Ω ′ Γ ≤ ξ and | I λ ( u ) − c Γ | ≤ ξ o . (6.1)We observe that w ∈ A λξ ∩ I c Γ λ , showing that A λξ ∩ I c Γ λ = ∅ . We have the following uniform estimate of (cid:13)(cid:13) I ′ λ ( u ) (cid:13)(cid:13) E ∗ λ on the region (cid:0) A λ ξ \ A λξ (cid:1) ∩ I c Γ λ . Proposition 6.1.
For each ξ > , there exist Λ ∗ ≥ and σ > independent of λ such that (cid:13)(cid:13) I ′ λ ( u ) (cid:13)(cid:13) E ∗ λ ≥ σ , for λ ≥ Λ ∗ and u ∈ (cid:0) A λ ξ \ A λξ (cid:1) ∩ I c Γ λ . (6.2) Proof.
We assume that there exist λ n → ∞ and u n ∈ (cid:16) A λ n ξ \ A λ n ξ (cid:17) ∩ I c Γ λ n such that (cid:13)(cid:13) I ′ λ n ( u n ) (cid:13)(cid:13) E ∗ λn → . Since u n ∈ A λ n ξ , we know ( k u n k λ n ) and (cid:0) I λ n ( u n ) (cid:1) are both bounded. Passing to a subsequenceif necessary, we may assume that ( I λ n ( u n )) converges. Thus, from Proposition 4.1, there exists u ∈ H (Ω Γ ) such that u is a solution for − ∆ u + u = (cid:16) Z Ω Γ | u | p | x − y | µ dy (cid:17) | u | p − u in Ω Γ with u n → u in H ( R ) , k u n k λ n , R \ Ω → and I λ n ( u n ) → I Γ ( u ) . As ( u n ) ⊂ Θ δ , we derive that k u n k λ n , Ω ′ j > rτ , j = 1 , · · · , l. Letting n → + ∞ , we get the inequality k u k j ≥ rτ > , j = 1 , · · · , l, which yields u | Ω j = 0 , j = 1 , · · · , l and I ′ Γ ( u ) = 0 . Consequently, I Γ ( u ) ≥ c Γ . However, from thefact that I λ n ( u n ) ≤ c Γ and I λ n ( u n ) → I Γ ( u ) , we derive that I Γ ( u ) = c Γ , and so, u ∈ Υ Γ . Thus,for n large enough k u n k j > rτ and | I λ n ( u n ) − c Γ | ≤ ξ, j = 1 , · · · , l. So u n ∈ A λ n ξ , which is a contradiction, finishing the proof.20n the sequel, ξ , ξ ∗ will be defined as ξ = min ( t , ··· ,t l ) ∈ ∂ [ r,R ] l | I Γ ( γ ( t , · · · , t l )) − c Γ | > and ξ ∗ = min { ξ / , δ, ρ/ } , where δ was given in(6.1) and ρ = 4 R c Γ , where R was fixed in (5.2). Moreover, for each s > , B λs denotes the set B λs = (cid:8) u ∈ E λ ; k u k λ ≤ s (cid:9) for s > . Proposition 6.2.
Suppose that < µ < and < p < − µ . Let ξ ∈ (0 , ξ ∗ ) and Λ ∗ ≥ given in the previous proposition. Then, for λ ≥ Λ ∗ , there exists a solution u λ of ( C λ ) such that u λ ∈ A λξ ∩ I c Γ λ ∩ B λ ρ +1 .Proof. Let λ ≥ Λ ∗ . Assume that there are no critical points of I λ in A λξ ∩ I c Γ λ ∩ B λ ρ +1 . Since I λ verifies the ( P S ) d condition with < d ≤ c Γ , there exists a constant ν λ > such that (cid:13)(cid:13) I ′ λ ( u ) (cid:13)(cid:13) E ∗ λ ≥ ν λ , for all u ∈ A λξ ∩ I c Γ λ ∩ B λ ρ +1 . From Proposition 6.1, we have (cid:13)(cid:13) I ′ λ ( u ) (cid:13)(cid:13) E ∗ λ ≥ σ , for all u ∈ (cid:0) A λ ξ \ A λξ (cid:1) ∩ I c Γ λ , where σ > is small enough and it does not depend on λ . In what follows, Ψ : E λ → R is acontinuous functional verifying Ψ( u ) = 1 , for u ∈ A λ ξ ∩ Θ δ ∩ B λ ρ , Ψ( u ) = 0 , for u / ∈ A λ ξ ∩ Θ δ ∩ B λ ρ +1 and ≤ Ψ( u ) ≤ , ∀ u ∈ E λ . We also consider H : I c Γ λ → E λ given by H ( u ) = ( − Ψ( u ) (cid:13)(cid:13) Y ( u ) (cid:13)(cid:13) − Y ( u ) , for u ∈ A λ ξ ∩ B λ ρ +1 , , for u / ∈ A λ ξ ∩ B λ ρ +1 , where Y is a pseudo-gradient vector field for I λ on K = { u ∈ E λ ; I ′ λ ( u ) = 0 } . Observe that H iswell defined, once I ′ λ ( u ) = 0 , for u ∈ A λ ξ ∩ I c Γ λ . The inequality (cid:13)(cid:13) H ( u ) (cid:13)(cid:13) ≤ , ∀ λ ≥ Λ ∗ and u ∈ I c Γ λ , guarantees that the deformation flow η : [0 , ∞ ) × I c Γ λ → I c Γ λ defined by dηdt = H ( η ) , η (0 , u ) = u ∈ I c Γ λ verifies ddt I λ (cid:0) η ( t, u ) (cid:1) ≤ − Ψ (cid:0) η ( t, u ) (cid:1)(cid:13)(cid:13) I ′ λ (cid:0) η ( t, u ) (cid:1)(cid:13)(cid:13) ≤ , (6.3) (cid:13)(cid:13)(cid:13)(cid:13) dηdt (cid:13)(cid:13)(cid:13)(cid:13) λ = (cid:13)(cid:13) H ( η ) (cid:13)(cid:13) λ ≤ (6.4)21nd η ( t, u ) = u for all t ≥ and u ∈ I c Γ λ \ A λ ξ ∩ B λ ρ +1 . (6.5)We study now two paths, which are relevant for what follows: • The path ( t , · · · , t l ) η (cid:0) t, γ ( t , · · · , t l ) (cid:1) , where ( t , · · · , t l ) ∈ [ r, R ] l .Since ξ ∈ (0 , ξ ∗ ) , we have that γ ( t , · · · , t l ) / ∈ A λ ξ , ∀ ( t , · · · , t l ) ∈ ∂ [ r, R ] l . and I λ (cid:0) γ ( t , · · · , t l ) (cid:1) < c Γ , ∀ ( t , · · · , t l ) ∈ ∂ [ r, R ] l . Once γ ( t , · · · , t l ) ∈ Θ δ , for all ( t , · · · , t l ) ∈ [ r, R ] l , (6.5) gives that η (cid:0) t, γ ( t , · · · , t l ) (cid:1) | Ω ′ j = 0 , t ≥ . Moreover, it is also easy to see that Z R \ Ω ′ (cid:0) |∇ η (cid:0) t, γ (cid:1) | + ( λa ( x ) + 1) | η (cid:0) t, γ (cid:1) | (cid:1) dx − p Z R \ Ω ′ (cid:18) | x | µ ∗ | η (cid:0) t, γ (cid:1) | p (cid:19) | η (cid:0) t, γ (cid:1) | p dx ≥ . Consequently, η (cid:0) t, γ ( t , · · · , t l ) (cid:1) ∈ Γ ∗ , t ≥ . • The path ( t , · · · , t l ) γ ( t , · · · , t l ) , where ( t , · · · , t l ) ∈ [ r, R ] l .We observe that supp (cid:0) γ ( t , · · · , t l ) (cid:1) ⊂ Ω Γ and I λ (cid:0) γ ( t , · · · , t l ) (cid:1) does not depend on λ ≥ , for all ( t , · · · , t l ) ∈ [ r, R ] l . Moreover, I λ (cid:0) γ ( t , · · · , t l ) (cid:1) ≤ c Γ , ∀ ( t , · · · , t l ) ∈ [ r, R ] l and I λ (cid:0) γ ( t , · · · , t l ) (cid:1) = c Γ if, and only if, t j = 1 , j = 1 , · · · , l. Therefore m = sup (cid:8) I λ ( u ) ; u ∈ γ (cid:0) [ r, R ] l (cid:1) \ A λξ (cid:9) is independent of λ and m < c Γ . In the following, we suppose that there exists K ∗ > such that (cid:12)(cid:12) I λ ( u ) − I λ ( v ) (cid:12)(cid:12) ≤ K ∗ k u − v k λ , ∀ u, v ∈ B λ ρ . Now, we will prove that max ( t , ··· ,t l ) ∈ [ r,R ] l I λ (cid:16) η (cid:0) T, γ ( t , · · · , t l ) (cid:1)(cid:17) ≤ c Γ − σ ξ K ∗ , (6.6)for T > large. In fact, writing u = γ ( t , · · · , t l ) , ( t , · · · , t l ) ∈ [ r, R ] l , if u / ∈ A λξ , from (6.3), wededuce that I λ (cid:0) η ( t, u ) (cid:1) ≤ I λ ( u ) ≤ m , ∀ t ≥ , u ∈ A λξ and set e η ( t ) = η ( t, u ) , f ν λ = min { ν λ , σ } and T = σ ξK ∗ f ν λ . Now, we will discuss two cases:
Case 1: e η ( t ) ∈ A λ ξ ∩ Θ δ ∩ B λ ρ , ∀ t ∈ [0 , T ] . Case 2: e η ( t ) / ∈ A λ ξ ∩ Θ δ ∩ B λ ρ , for some t ∈ [0 , T ] . Analysis of Case 1
In this case, we have Ψ (cid:0)e η ( t ) (cid:1) = 1 and (cid:13)(cid:13) I ′ λ (cid:0)e η ( t ) (cid:1)(cid:13)(cid:13) ≥ f ν λ for all t ∈ [0 , T ] . Hence, from (6.3), weknow I λ (cid:0)e η ( T ) (cid:1) = I λ ( u ) + Z T dds I λ (cid:0)e η ( s ) (cid:1) ds ≤ c Γ − Z T f ν λ ds, that is, I λ (cid:0)e η ( T ) (cid:1) ≤ c Γ − f ν λ T ≤ c Γ − σ ξ K ∗ , showing (6.6). Analysis of Case 2 : In this case we have the following situations: (a) : There exists T ∈ [0 , T ] such that ˜ η ( t ) / ∈ Θ δ . Let T = 0 it follows that k ˜ η ( T ) − ˜ η ( T ) k ≥ δ > µ, because ˜ η ( T ) = u ∈ Θ . (b) : There exists T ∈ [0 , T ] such that ˜ η ( T ) / ∈ B λ ρ . Let T = 0 , we get k ˜ η ( T ) − ˜ η ( T ) k ≥ ρ > µ, since ˜ η ( T ) = u ∈ B λρ . (c) : ˜ η ( t ) ∈ Θ δ ∩ B λ ρ for all t ∈ [0 , T ] , and there are ≤ T ≤ T ≤ T such that ˜ η ( t ) ∈ A λ ξ \ A λξ for all t ∈ [ T , T ] with | I λ (˜ η ( T )) − c Γ | = ξ and | I λ (˜ η ( T )) − c Γ | = 3 ξ From definition of K ∗ , we have k ˜ η ( T ) − ˜ η ( T ) k ≥ K ∗ (cid:12)(cid:12) I λ (˜ η ( T )) − I λ (˜ η ( T )) (cid:12)(cid:12) ≥ K ∗ ξ, then the mean value theorem implies that T − T ≥ K ∗ ξ . Notice that I λ (cid:0)e η ( T ) (cid:1) ≤ I λ ( u ) − Z T Ψ (cid:0)e η ( s ) (cid:1)(cid:13)(cid:13) I ′ λ (cid:0)e η ( s ) (cid:1)(cid:13)(cid:13) ds, we can deduce that I λ (cid:0)e η ( T ) (cid:1) ≤ c Γ − Z T T σ ds = c Γ − σ ( T − T ) ≤ c Γ − σ ξ K ∗ , b η ( t , · · · , t l ) = η (cid:0) T, γ ( t , · · · , t l ) (cid:1) , ( t , · · · , t l ) ∈ [ r, R ] l , we have that b η ∈ Γ ∗ and max ( t , ··· ,t l ) ∈ [ r,R ] l I λ (cid:0)b η ( t , · · · , t l ) (cid:1) ≤ c Γ − σ ξ K ∗ , ∀ λ ≥ Λ ∗ . On the other hand, we can estimate I λ (cid:0)b η (cid:1) = 12 Z R ( |∇ b η | + ( λa ( x ) + 1) | b η | ) dx − p Z R (cid:16) | x | µ ∗ | b η | p (cid:17) | b η | p dx = 12 Z Ω ′ Γ ( |∇ b η | + ( λa ( x ) + 1) | b η | ) dx − p Z Ω ′ Γ (cid:16) Z Ω ′ Γ | b η | p | x − y | µ dy (cid:17) | b η | p dx + 12 Z R \ Ω ′ Γ ( |∇ b η | + ( λa ( x ) + 1) | b η | ) dx − p Z R \ Ω ′ Γ (cid:16) | x | µ ∗ | b η | p (cid:17) | b η | p dx − p Z Ω ′ Γ (cid:16) Z R \ Ω ′ Γ | b η | p | x − y | µ dy (cid:17) | b η | p dx ≥ I λ, Γ ( b η ) + 12 Z R \ Ω ′ Γ ( |∇ b η | + ( λa ( x ) + 1) | b η | ) dx − p Z R \ Ω ′ Γ (cid:16) | x | µ ∗ | b η | p (cid:17) | b η | p dx Since b η ∈ Γ ∗ , it follows that Z R \ Ω ′ Γ ( |∇ b η | + ( λa ( x ) + 1) | b η | ) dx − p Z R \ Ω ′ Γ (cid:16) | x | µ ∗ | b η | p ) (cid:17) | b η | p dx ≥ , and so, I λ (cid:0)b η (cid:1) ≥ I λ, Γ ( b η ) . (6.7)By (6.6) and (6.7), applying Lemma 5.2, we have c λ, Γ ≤ max (cid:26) m , c Γ − σ ξ K ∗ (cid:27) ∀ λ ≥ Λ ∗ , which leads to lim sup λ → + ∞ c λ, Γ ≤ max (cid:26) m , c Γ − σ ξ K ∗ (cid:27) < c Γ , this contradicts with the conclusion ( ii ) of Lemma 5.1. [Proof of Theorem 1.2: Conclusion] From the last Proposition, there exists ( u λ n ) with λ n → + ∞ satisfying:(a) I ′ λ n ( u λ n ) = 0 , ∀ n ∈ N ;(b) I λ n ( u λ n ) → c Γ . (c) || u λ n || λ n , R N \ Ω ′ Γ → . Therefore, from of Proposition 4.1, we derive that ( u λ n ) converges in H ( R ) to a function u ∈ H ( R ) , which satisfies u = 0 outside Ω and u | Ω j = 0 , j = 1 , · · · , l . Now, we claim that u = 0 in Ω j , for all j / ∈ Γ . Indeed, it is possible to prove that there is σ > , which is independentof j , such that if v is a nontrivial solution of ( C ) ∞ , Γ , then k v k H (Ω Γ ) ≥ σ . u verifies k u k H ( R N \ Ω Γ ) = 0 , showing that u = 0 in Ω j , for all j / ∈ Γ . This finishes the proof of Theorem 1.2. ACKNOWLEDGMENTS
The authors would like to thank the anonymous referee for his/heruseful comments and suggestions which help to improve and clarify the paper greatly.
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