STATIONARY SOLUTIONS OF THE NONLINEAR SCHRÖDINGER EQUATION WITH FAST-DECAY POTENTIALS CONCENTRATING AROUND LOCAL MAXIMA JONATHANDICOSMOANDJEANVANSCHAFTINGEN 2 Abstract. Westudypositiveboundstatesfortheequation 1 −ε2∆u+Vu=up, inRN, 0 where ε > 0 is a real parameter, N < p < N+2 and V is a nonnegative 2 N−2 N−2 potential. Using purely variational techniques, we find solutions which con- n centrate at local maxima of the potential V without any restriction on the a potential. J 5 2 ] 1. Introduction P A We consider the stationary nonlinear Schrödinger equation . h −ε2∆u+Vu=up, in RN, t (P ) a ε ( u(x)→0 as |x|→∞, m [ where ε>0 is a real parameter, N ≥3, N 0. In the critical frequency case infRN V = 0, solutions concentrating i X around nondegenerate critical points [5] and around local minima have been ob- r tained [3,10] provided that the potential V does not decay too fast at infinity. In a the case of local minima, the variational method has been adapted to construct solutions concentrating around a local minimum with a fast decay potential V — including a compactly supported potential [6,20,21,28]. The goal of this work is to establish by a variational method the existence of solutionsconcentratingaroundlocalmaximaforfast-decayingpotentials. Sinceany potentialthat decaysatinfinity has a globalmaximum, this showsthe existence of solutions for a quite generalclass of potentials. We also think that this problem is agoodtestoftherobustnessandflexibilityofthevariationalmethodsforsolutions Date:January 26,2012. 2010 Mathematics Subject Classification. 35J65(35B05, 35B25,35B40,35J20, 35Q55). Keywordsandphrases. StationarynonlinearSchrödingerequation;semiclassicalstates;semi- linearellipticproblem;vanishingpotential;criticalfrequency;concentrationaroundlocalmaxima. JonathanDiCosmoisaresearch fellowoftheFondsdelaRecherche Scientifique–FNRS. 1 2 JONATHANDICOSMOANDJEANVANSCHAFTINGEN concentrating around local maxima and of the penalization method for fast decay potentials. Our main result is the following Theorem 1. Let N ≥ 1, p > 1 such that 1 > N−2 and V ∈ CN(RN,R+), p N+2 V 6≡ 0 be a nonnegative potential. If lim V(x) = 0 and either 1 < N−2 or |x|→∞ p N liminf V(x)|x|2 > 0, then, for ε > 0 small enough, the problem (P ) has a |x|→∞ ε family of positive solution that concentrates around a global maximum of V. Atypicalnew potentialV forwhichthis resultappliesisgivenbyV(x)= 1 1+|x|4 for x ∈ Rn. The assumption on p is optimal, since when 1 ≥ N−2 and V is p N compactly supported, (P ) does not have any solution. Indeed, such a solution ε would be positive and satisfy −∆u = up in RN \suppV and that would imply u=0 on RN \suppV [7]. Theorem 1 follows from the following result: Theorem 2. Let N ≥ 1, p> 1 such that 1 > N−2 and V ∈C(RN,R+). Assume p N+2 that there exists a smooth bounded open set Λ⊂RN such that supV >infV =supV Λ Λ ∂Λ and supV pp−+11−N2 <2infV pp−+11−N2 . Λ Λ If either 1 < N−2 or liminf V(x)|x|2 > 0, then for ε > 0 small enough, p N |x|→∞ problem (P )possessesapositiveweaksolutionu ∈H1 (RN)suchthatu achieves ε ε loc ε its maximum at x ∈Λ, ε liminfu (x )>0 ε ε ε→0 and liminfdist(x ,RN \Λ)>0. ε ε→0 The first assumption on V implies that supV =infV , ∂Λ ∂Λ that is, ∂Λ is a level line of V. Theorem1followsfromTheorem2bytakingΛ ={x∈RN | V(x)>supV−δ} δ for δ > 0. By Sard’s lemma, the set Λ is smooth for almost every δ > 0. One δ applies then Theorem 2 and a diagonal argument. Our method of proof is based on the penalization method [11,12] adapted to decayingpotentials[10,21]. Howeverthedecayofthe potentialsrequiresustotake some extra care at several steps, especially when lower bounds on the energy of solutions are needed. This paper is organized as follows. We first introduce a penalized problem (section 2) and recall some properties of the associated limiting problem (section 3). We then study the asymptotic behaviour of families of critical points (section 4) and minimizers (section 5) ofthe energyfunctional associatedto (P ). This allows ε us to define a minimax level and prove the existence of a solution to the penalized probleminsection6. Finallyinsection7weusetheasymptoticsandsomecompar- isonargumentto showthatwhen εis small,oursolutionsofthe penalizedproblem solvethe originalproblem. Whereasthe proofis writtenfor N ≥3,we highlightin section 8 how the proof can be adapted to the case N ≤2. STATIONARY SOLUTIONS AROUND LOCAL MAXIMA 3 2. The penalized problem Following M.del Pino and P.Felmer [12], we introduce a penalized problem. D.BonheureandJ.VanSchaftingen[9,10]haveintroducedapenalizedproblemfor decaying potentials. The penalization for fast decay potentials is due to V.Moroz and J.Van Schaftingen [20,21]. It was used by D.Bonheure together with the au- thors to study solutions concentrating aroundspheres [8]. Another penalized problem was defined by Yin Huicheng and Zhang Pingzheng [28] (see also Fei Mingwen and Yin Huicheng [15] and Ba Na, Deng Yinbin and Peng Shuangjie [6]). 2.1. Thepenalizationpotential. RecallthatΛisaboundeddomain. Letx ∈Λ 0 and ρ>0 be such that B(x ,ρ)⊂Λ, and let χ denote the characteristicfunction 0 Λ of the set Λ. For N ≥3, the penalization potential H :RN →R is defined by (N −2)2 log ρ 1+β H(x):= 1−χ (x) ρ0 Λ 4|x−x |2 log|x−x0| (cid:0) (cid:1) 0 (cid:18) ρ0 (cid:19) for some fixed β >0 and ρ ∈(0,ρ). 0 Letusrecallthattheoperator−∆−H satisfiesapositivityprinciple[21,Lemma 3.1]. Lemma 2.1. For every u∈C∞(RN), c |∇u|2−H|u|2 ≥0. ZRN (cid:0) (cid:1) Proof. Since N ≥3, this follows fromthe classicalHardy inequality since for every x∈RN \B(x ,ρ), 0 (N −2)2 H(x)≤ . (cid:3) 4|x−x |2 0 2.2. The penalized nonlinearity. Fix µ ∈ (0,1). The penalized nonlinearity g :RN ×R→R is defined for x∈RN and s∈R by ε g (x,s):=χ (x)s p+ 1−χ (x) min µ ε2H(x)+V(x) ,|s|p−1 s . ε Λ + Λ + Also set G (x,s) := sg (x,(cid:0)σ)dσ. Th(cid:1)e fun(cid:0)cti(cid:0)on g is a Cara(cid:1)théodor(cid:1)y function ε 0 ε ε with the following properties : R (g ) g (x,s)=o(s), as s→0, uniformly on compact subsets of RN. 1 ε (g ) for every x∈RN and s∈R, 2 g (x,s)≤(s)p, ε + if moreover x∈RN \Λ, then g (x,s)≤µ ε2H(x)+V(x) s ε + (g ) for every s∈R, if x∈Λ, (cid:0) (cid:1) 3 (p+1)G (x,s)≤g (x,s)s, ε ε and if x6∈Λ, 2G (x,s)≤g (x,s)s, ε ε (g ) the function 4 g (x,ts)s ε t∈(0,∞)7→ t is nondecreasing for all x∈RN and s∈R. 4 JONATHANDICOSMOANDJEANVANSCHAFTINGEN 2.3. The penalized functional. The Hilbert space naturally associated to the linear part of our equation is the weighted Sobolev space H1(RN), which is the V closure of C∞(Rn) under any of the equivalent norms c kuk2 := ε2|∇u|2+V |u|2 ε ZRN defined for ε>0. (cid:0) (cid:1) We look for a solution u∈H1(RN) of the penalized equation V (Q ) −ε2∆u(x)+V(x)u(x)=g x,u(x) for x∈RN. ε ε The associated functional is (cid:0) (cid:1) 1 J :H1 →R:J (u):= ε2|∇u|2+V |u|2 − G x,u(x) dx. ε V ε 2ZRN ZRN ε It is standard that J is well-define(cid:0)d and continuous(cid:1)ly different(cid:0)iable an(cid:1)d that its ε critical points are weak solutions of the penalized equation (Q ). ε 2.4. The Nehari manifold. The Neharimanifoldassociatedtothe functional J ε is defined by N := u∈H1(RN)\{0} | hJ′(u),ui=0 . ε V ε Itiswell-knownthatu∈H1(RN)\{0}isacriticalpointofJ ifandonlyifu∈N (cid:8) V ε(cid:9) ε and u is a critical point of J restricted to N . ε ε WepointoutthatN isboundedawayfrom0. Wefirsthaveanintegralestimate. ε Lemma 2.2. Let ε>0 and u∈N . Then ε (u)p+1 ≥(1−µ) ε2|∇u|2+V |u|2. + ZΛ ZRN Proof. By (g ), one has 2 ε2|∇u|2+V |u|2 = g (x,u(x))u(x)dx ε ZRN ZRN (cid:0) (cid:1) ≤ |u|p+1+µ (V +ε2H)|u|2. ZΛ ZRN\Λ We deduce from Lemma 2.1 that (1−µ) ε2|∇u|2+V |u|2 ≤ |u|p+1. (cid:3) ZRN ZΛ Lemma 2.3. Let ε>0 and u∈N . Then ε ε2|∇u|2+V |u|2 ≥cεN ZΛ where c>0 is independent of ε and u. Proof. Since inf V >0, the Sobolev and Hölder inequalities imply that Λ p−1N p+1−p−1N |u|p+1 ≤C |∇u|2 4 |u|2 2 4 ZΛ (cid:16)ZRN (cid:17) (cid:16)ZΛ (cid:17) C p+1 ≤ ε2|∇u|2+V |u|2 2 . The conclusion follows fromεLpe−2m1Nm(cid:16)aZ2Λ.2. (cid:17) (cid:3) We also have a uniform lower estimate on the maximum. Lemma 2.4. Let ε>0 and u∈N . Then ε up−1 sup + ≥1. V Λ STATIONARY SOLUTIONS AROUND LOCAL MAXIMA 5 This was proved for solutions of (Q ) by V.Moroz and J.Van Schaftingen [21, ε Lemma 4.2] (see also [10, Lemma 17]). Proof. One has by (g ), 2 ε2|∇u|2+V |u|2 ≤ (u)p+1+µ (V +ε2H)|u|2 + ZRN ZΛ ZRN\Λ up−1 ≤sup + V |u|2+µ (V +ε2H)|u|2 , Λ V ZΛ ZRN\Λ and thus by Lemma 2.1, up−1 sup + V |u|2 ≥ V |u|2 . V Λ ZΛ ZΛ By Lemma 2.3, V |u|2 >0, and the conclusion follows. (cid:3) Λ R We also note the following coercivity estimate. Lemma 2.5. For every ε>0 and u∈N , ε 1 1 − (1−µ) ε2|∇u|2+V |u|2 ≤J (u). ε (cid:16)2 p+1(cid:17) ZRN(cid:0) (cid:1) Proof. Since u∈N , one has ε 1 1 J (u)= − ε2|∇u|2+V |u|2 ε (cid:16)2 p+1(cid:17)ZRN(cid:0) 1 (cid:1) + g x,u(x) u(x)−(p+1)G x,u(x) dx. ε ε p+1ZRN (cid:0) (cid:1) (cid:0) (cid:1) In view of (g ) and (g ), 3 2 1 g x,u(x) u(x)−(p+1)G (x,u(x))dx ε ε p+1ZRN (cid:0) (cid:1) p−1 ≥− G x,u(x) dx ε p+1ZRN\Λ 1(cid:0) 1(cid:1) ≥− − µ (ε2H +V)|u|2. (cid:16)2 p+1(cid:17) ZRN\Λ Thanks to Lemma 2.1, we reach the conclusion. (cid:3) 2.5. The Palais-Smale condition. For every ε > 0, the functional J satisfies ε the Palais-Smale compactness condition: Lemma 2.6. For every ε > 0, if (un)n∈N is a sequence such that (Jε(un))n∈N converges and (Jε′(un))n∈N converges to0 in(HV1(RN))′, then, uptoa subsequence, (un)n∈N converges in HV1(RN). The proof of Lemma 2.6 is a combination of the arguments for the penalization withoutH [10,Lemma6]andwithoutV [21,Lemma3.5]whosemainlinesoriginate intheprooffornondecayingpotentials[11,Lemma1.1]. Itwasalreadyprovedwith the present penalization for the functional restricted to a subspace of symmetric functions [8]. 6 JONATHANDICOSMOANDJEANVANSCHAFTINGEN 2.6. Minimizers on the Nehari manifold. Proposition 2.7. For every ε>0, there exists u∈N such that ε J (u)=infJ . ε ε Nε Proposition 2.7 was proved for the penalization for nondecaying potentials [11, Lemma2.1],thepenalizationwithoutH [10,Proposition9]thepenalizationwithout V [10, Proposition 3.7] and the present penalization under symmetry constraints [8]. Proof of Proposition 2.7. The proof is standard: by (g ) one has the equality [25, 4 Proposition 3.11] infJ = inf supJ (tu)= inf sup J (γ(t)); ε ε ε Nε u∈HV1(RN)t>0 γ∈C([0,1],HV1(RN))t∈[0,1] u+|Λ6=0 γ(0)=0 Jε(γ(1))<0 by Lemmas 2.3 and 2.5, J is bounded away from 0 on N . Since J satisfies the ε ε ε Palais-Smale compactness condition by Lemma 2.6, the existence of u follows. (cid:3) 3. Limiting problems 3.1. The limit problem. For ν >0 let U be the unique positive solution of the ν problem −∆u+νu=up in RN, (3.1)  u>0,  u(0)=maxu. RN The function Uν is radialaroundthe origin[17]. The functional associatedto (3.1) is I :H1(RN)→R defined for u∈H1(RN) by ν 1 1 I (u):= |∇u|2+ν|u|2 − up+1. ν 2ZRN p+1ZRN + (cid:0) (cid:1) One has the variational characterization b := inf I =I (U ) ν ν ν ν Mν where M = u∈H1(RN)\{0} | |∇u|2+ν|u|2 = up+1 . ν + n ZRN ZRN o (cid:0) (cid:1) We also set Sr (3.2) C(y):=bV(y) = pr+1V(y)pp+−11−N2, where 1 = 1 − 1 and r 2 p+1 S2 :=inf |∇u|2+|u|2 | u∈H1(RN) and up+1 =1 . p+1 + nZRN(cid:0) (cid:1) ZRN o We also recall the following classical result Lemma 3.1. Let ν >0 and (vn)n∈N be a sequence in Mν ⊂H1(RN). If lim I (v )=b , ν n ν n→∞ then there exists a sequence of points (yn)n∈N in RN such that vn(·−yn)→Uν in H1(RN). STATIONARY SOLUTIONS AROUND LOCAL MAXIMA 7 Proof. Let F :H1(RN)→R be defined for v ∈H1(RN) by ν F (v)= |∇v|2+ν|v|2− vp+1 ν + ZRN ZRN By a standard application of the Ekeland variational principle on the manifold Mν (see for example [19, Theorem 4.1]), there exist sequences (vñ)n∈N ⊂Mν and (λn)n∈N ⊂ R such that Iν(vñ) → bν, Iν′(vñ)+λnFν′(vñ) → 0 in H−1(RN) and vn − vñ → 0 in H1(RN) as n → ∞. The sequence (vñ)n∈N is a Palais-Smale sequence for the unconstrained functional I , that is I′(v˜ )→0. Indeed one has ν ν n λ (p−1) |∇v |2+|v |2 =−λ hF′(v˜ ),v˜ i n n n n ν n n ZRN =hI′(v˜ ),v˜ i+o(kv˜ k )=o(kv˜ k ), ν n n n H1 n H1 as n → ∞. Since there exists a constant c > 0 such that kvk ≥ c for every H1(RN) v ∈M we deduce that lim λ =0. ν n→∞ n We compute that 2 2I (v˜ )−hI′(v˜ ),v˜ i= 1− (v˜ )p+1dx→2b . ν n ν n n (cid:16) p+1(cid:17)ZRN n + ν Hence, liminf (v˜ )p+1 >0. n + n→∞ ZRN Since (vñ)n∈N is bounded in H1(RN), we deduce from[18, Part2, Lemma I.1] (see also [26, Lemma 1.21]) that p−1 (v˜ )p+1 ≤C |∇v˜ |2+|v˜ |2 sup (v˜ )p+1 p+1 . n + n n n ZRN (cid:16)ZRN (cid:17)z∈RN(cid:16)ZB(z,1) (cid:17) Consequently, there exists a sequence (yn)n∈N ⊂ RN such that, if we set v¯n := v˜ (·−y ), we have n n (3.3) liminf (v¯ )p+1 >0. n + n→∞ ZB(0,1) Since (v¯n)n∈N is bounded in H1(RN) and p1 > NN−2, we can assume that v¯n ⇀ v¯ in H1(RN), v¯ →v¯ in Lp+1(RN) and v¯ →v¯ almost everywhere. By (3.3), v¯6≡ 0. n loc n Forallv ∈H1(RN),wehavehI′(v¯ ),vi→0becausev¯ isaPalais-Smalesequence, ν n n and hI′(v¯ ),vi→hI′(v¯),vi because v¯ ⇀v¯. We conclude that hI′(v¯),vi=0 and ν n ν n ν so v¯ is a solution of (3.1). We compute that p−1 (v¯ )p+1 =2I (v¯ )−hI′(v¯ ),v¯ i→2b p+1ZRN n + ν n ν n n ν and p−1 (v¯)p+1 =2I (v¯)=2b . p+1ZRN + ν ν Therefore k(v¯ ) k →k(v¯) k . We infer that v¯ →v¯ in Lp+1(RN). n + Lp+1(RN) + Lp+1(RN) n Finally we can write kv¯ −v¯k2 =hI′(v¯ )−I′(v¯),v¯ −v¯i n H1(RN) ν n ν n + (v¯ )p −(v¯)p (v¯ −v¯) . n + + n ZRN (cid:0) (cid:1) Since I′(v¯) = 0, I′(v¯ ) → 0 as n → ∞ and the last term goes to 0 by Hölder’s ν ν n inequality, we conclude that v¯ →v¯ in H1(RN). The conclusion follows. (cid:3) n 8 JONATHANDICOSMOANDJEANVANSCHAFTINGEN 3.2. Penalized limit problems. The two following lemmas will provide informa- tion about the limit of sequences of rescaled solutions. The first lemma is due to M.del Pino and P.Felmer [13, Lemma 2.3]. Let RN := x∈RN | x >0 . + N Lemma 3.2. Let ν ≥0 and µ∈[0,ν]. If u∈H1(RN) (cid:8)is a solution of (cid:9) −∆u+νu=up in RN, + + (−∆u+νu=min µ,|u|p−1 u+ in RN−, then |u|p−1 ≤µ in RN. (cid:0) (cid:1) − Proof. We follow the argument of M.del Pino and P.Felmer [13, Lemma 2.3]. By elliptic regularity,u∈H2(RN)∩C1(RN). Thus we can use ∂ u as a test function N in the equation. Writing g(s) := min(µ,|s|p−1)s and G(s) := sg(σ) dσ, we + 0 obtain R 1 1 ∂ |∇u|2+ν|u|2 = ∂ up+1 + ∂ (G◦u). 2ZRN N p+1ZRN+ N + ZRN− N (cid:0) (cid:1) (cid:0) (cid:1) This reduces to G u(x′,0) − 1 u(x′,0) p+1 dx′ =0. ZRN−1(cid:16) (cid:0) (cid:1) p+1(cid:0) (cid:1)+ (cid:17) SinceG(u)≤ up++1 onRN,wehaveG(u(x′,0))= 1 (u(x′,0))p+1 forallx′ ∈RN−1 p+1 p+1 + and hence, for every x′ ∈RN−1, u(x′,0)≤µp−11 . One has on RN − −∆u+(ν−µ)u≤0. Since ν ≥µ, we deduce by the maximum principle that u≤µp−11 in RN. (cid:3) − The second lemma is an application of the maximum principle. Lemma 3.3. Let ν ≥0 and µ∈[0,ν]. If u∈H1(RN), u≥0, is a solution of −∆u+νu=min µ,up−1 u in RN, + then u≡0. (cid:0) (cid:1) Proof. If u is a solution, we have −∆u+(ν−µ)u≤0 in RN . Taking u as a test function, we obtain |∇u|2+(ν−µ)|u|2 ≤0. ZRN Since ν−µ≥0, this implies that u≡0. (cid:3) 4. Asymptotics of families of critical points In this section we refine the asymptotic analysis in [10, Section 5] in order to obtain an estimate of the energy of a critical point u of J depending on the ε ε number and the location of its local maxima. The corresponding lower estimate was proved in [10]. STATIONARY SOLUTIONS AROUND LOCAL MAXIMA 9 4.1. Asymptotics on small balls. The next lemma states that the sequences of rescaled solutions converge in C1 (RN) to a function in H1(RN). loc Lemma 4.1. Let (εn)n∈N be a sequence in R+ such that εn → 0 as n → ∞, let (un)n∈N be a sequence of solutions of Qεn such that liminfε−NJ (u )<∞ n→∞ n εn n and let (xn)n∈N be a sequence in RN such that xn → x¯ as n → ∞. Denote by (vn)n∈N the sequence defined by vn(x) = uεn(xn +εnx). If V(x¯) > 0, then there exists v ∈H1(RN) such that, up to a subsequence, v →v in C1 (RN) as n→∞. n loc This lemma was proved for minimal energy solutions in [10, Lemma 13]. We sketch here the argument in order to highlight that the proof only depends on the fact that u is a solution that satisfies an energy bound. n Proof. Take ϕ ∈ C∞(RN) such that ϕ ≡ 1 on B(0,1). Set for R > 0 and x ∈ RN c ϕR(x) = ϕ(Rx). The sequence (ϕRvn)n∈N is bounded in H1(RN) for every R > 0. By a diagonal argument, there exists v ∈ H1 (RN) such that v → v weakly in loc n H1 (RN) along a subsequence. loc Now note that for every R>0, |∇v|2 ≤liminf |∇v |2 ≤liminf |∇v |2 n n ZB(0,R) n→∞ ZB(0,R) n→∞ ZRN and V(x¯) |v|2 ≤liminf V |v |2 ≤liminf V |v |2 , n n ZB(0,R) n→∞ ZB(0,R) n→∞ ZRN so that v ∈H1(RN). The remainder follows from classical regularity and compactness results. (cid:3) Lemma 4.2. Let (εn)n∈N be a sequence in R+ such that εn → 0 as n → ∞, let (un)n∈N be a sequence of solutions of Qεn such that liminfε−NJ (u )<∞ n→∞ n εn n and let (xn)n∈N be a sequence in RN such that xn →x¯ as n→∞. If V(x¯)>0 and liminfu (x )>0, n→∞ εn n then, x¯∈Λ¯, dist(x ,Λ) n limsup <∞, ε n→∞ n 1 lim limsup ε−N ε2 |∇u |2+V |u |2 −G (.,u ) −C(x¯) =0, Ran→d∞ n→∞ (cid:12)(cid:12) n ZB(xn,εnR)(cid:16)2(cid:0) n εn εn (cid:1) εn εn (cid:17) (cid:12)(cid:12) (cid:12) (cid:12) lim limsupε−N |∇u |2+V |u |2 =0. R→∞ n→∞ n ZB(xn,2εnR)\B(xn,εnR) εn εn Proof. Setv (x):=u (x +ε x). ByLemma4.1uptoasubsequence,thereexists n εn n n a v ∈ H1(RN) such that v → v in C1 (RN). We have v(0) = lim v (0) > 0 n loc n→∞ n so that v 6≡0. Let us now prove by contradiction that dist(x ,Λ) n (4.1) limsup <∞. ε n→∞ n 10 JONATHANDICOSMOANDJEANVANSCHAFTINGEN Uptoasubsequence,wecanassumethatlim dist(x ,Λ)/ε =∞. Then,since n→∞ n n the sequenceofcharacteristicfunctions χ (x):=χ (x +ε x) convergespointwise n Λ n n to 0, we have as n→∞ g (x +ε ·,v )=min µ ε2H(x +ε ·)v +V(x +ε ·) ,|v |p−1 (v ) εn n n n n n n n n n n n + →(cid:0)m(cid:0)in µV(x¯),|v|p−1 v+, (cid:1) (cid:1) in Lq (RN) for 1≤q < 2N ,(cid:0)and thus v solv(cid:1)es the limiting equation loc p(N−2) −∆v+V(x¯)v =min µV(x¯),|v|p−1 v , in RN. + By Lemma 3.3, v ≡0, which is a con(cid:0)tradiction. Th(cid:1)us (4.1) holds. Now, let us assume that dist(x ,RN \Λ) n (4.2) limsup =∞. ε n→∞ n Since χ (x) converges pointwise to 1, we have, up to a subsequence, for n large n enough, g (x +ε ·,v )=(v )p →vp , εn n n n n + + in Lq (RN) for 1≤q < 2N . Hence v solves the limiting equation loc p(N−2) (4.3) −∆v+V(x¯)v =vp in RN . + If (4.1) holds but (4.2) does not, then dist(x ,∂Λ) n limsup <∞. ε n→∞ n SinceΛissmooth,χ →χ ,almosteverywhereasn→∞,whereE isahalf-space. n E By Lemma 3.2, v is again a solution of (4.3). In any case, v is thus a nontrivial solution of (4.3). Now we claim that 1 (4.4) lim lim ε−N ε2|∇u |2+V |u |2 R→∞n→∞ n 2ZB(xn,εnR) n εn εn (cid:0) 1 (cid:1) 2 2 = ∇U +V(x¯) U . 2ZRN V(x¯) V(x¯) For every R>0, the convergence of vn to v in Cl1oc(R(cid:12)(cid:12) N) imp(cid:12)(cid:12)lies that (cid:12)(cid:12) (cid:12)(cid:12) 1 1 lim ε−N ε2 |∇u |2+V |u |2 = |∇v|2+V(x¯)|v|2 . n→∞ n 2 n εn εn 2 ZB(xn,εnR) ZB(0,R) (cid:0) (cid:1) Sincev ∈H1(RN),andv andU areequaluptoatranslation,weconcludethat V(x¯) (4.4) holds. The argument for the other limit is similar. (cid:3) 4.2. Asymptotics outside small balls. The solutions decay outside a neighbor- hood of Λ: Lemma 4.3. For every open set U such that Λ¯ ⊂U, there exists C >0 such that for every ε>0, if u∈H1(RN) is a solution of Q , V ε ε2|∇u|2+V |u|2 ≤Cε2 ε2|∇u|2+V |u|2 . ZRN\U ZRN (cid:0) (cid:1) (cid:0) (cid:1) Proof. Since V is continuous and inf V > 0, we can assume without loss of gen- Λ erality that inf V > 0. Take ψ ∈ C∞(RN) such that ψ = 0 on Λ¯ and ψ = 1 on U c RN \U. By taking ψ2u as a test function in (Q ), we obtain n ε ε2|∇(ψu)|2+V |ψu|2 = g (x,u(x))ψ(x)2u(x)dx+ ε2|∇ψ|2|u|2. ε ZRN ZRN ZRN (cid:0) (cid:1)

Stationary solutions of the nonlinear Schrödinger equation with fast-decay potentials concentrating around local maxima

Similar Stationary solutions of the nonlinear Schrödinger equation with fast-decay potentials concentrating around local maxima