Concentration on circles for nonlinear Schrödinger-Poisson systems with unbounded potentials vanishing at infinity
aa r X i v : . [ m a t h . A P ] S e p CONCENTRATION ON CIRCLES FOR NONLINEARSCHRÖDINGER-POISSON SYSTEMS WITH UNBOUNDEDPOTENTIALS VANISHING AT INFINITY
DENIS BONHEURE, JONATHAN DI COSMO, AND CARLO MERCURI
Abstract.
The present paper is devoted to weighted Nonlinear Schrödinger- Pois-son systems with potentials possibly unbounded and vanishing at infinity. Using apurely variational approach, we prove the existence of solutions concentrating on acircle. Introduction
The aim of the present paper is to study of the behavior of a certain class of solutionsfor the following nonlinear Schrödinger-Poisson system − ε ∆ u + V ( x ) u + ρ ( x ) φu = K ( x ) u p , x ∈ R , − ∆ φ = ρ ( x ) u , (1)in the semiclassical limit, namely for ε → , where ε stands for the reduced Planck constant ~ . In particular, we focus on solutions concentrating on a circle.Let us choose any -dimensional linear subspace d ⊂ R . We denote by π the orthogonalcomplement of d . If x ∈ R , we will write x = ( x ′ , x ′′ ) with x ′ ∈ d and x ′′ ∈ π .As a particular case of our main result, we have the following theorem: Theorem 1.
Let p > and V ∈ C ( R \ { } , R + ) be a radial potential. Write V ( x ) =˜ V ( x ′ , | x ′′ | ) . If there exists r ∗ > such that the function M ( r ) := r h ˜ V (0 , r ) i p − has an isolated local minimum at r = r ∗ such that M ( r ∗ ) > , then for ε small enough,the system (cid:26) − ε ∆ u + V ( x ) u + φu = u p , x ∈ R , − ∆ φ = u , has a positive cylindrically symmetric solution u ε that concentrates on the circle of radius r ∗ centered at the origin and contained in the plane π . We point out that we have no assumption about the decay of V at infinity. In particular, V could be compactly supported. This is an improvement on previous works, see e.g. [1].A fundamental physical problem arises from the correspondence principle, accordingto which quantum mechanics contains classical mechanics as ~ → . In the framework ofthe Schrödinger equation with Coulomb potential one can construct solutions which arelocalized around classical Keplerian elliptic orbits by superposition of states of minimalquantum fluctuation (coherent states), see [9, 15]. Due to the dispersive nature of theSchrödinger equation, a rigorous reduction to classical mechanics cannot in general beperformed. By introducing a local nonlinear homogeneous term u p , in [4], the authorsprove the existence of solutions for the D nonlinear Schrödinger equation with radialpotential, concentrating on a circle. In the case of radial potentials, due to the invarianceby rotations, the classical and quantum angular momentum are conserved as indicated by Date : September 25, 2017.2000
Mathematics Subject Classification.
Key words and phrases.
Stationary nonlinear Schrödinger-Poisson system; weighted Sobolev spaces;degenerate potentials.Jonathan Di Cosmo is a research fellow of the Fonds de la Recherche Scientifique–FNRS.
Noether’s Theorem. This suggests that the solutions concentrating on Keplerian orbits aresuitable candidates in order to mimic, in the semi-classical limit, the classical dynamicsdescribed by Newton’s equations. In [8], the existence of solutions concentrating on circleshas been obtained for the D nonlinear Schrödinger equation with cylindrically symmetricpotential. In both [4, 8] the underlying idea is to find solutions with nonzero angularmomentum. By a different method in [7] and [5] the existence of solutions concentratingon points and, respectively, on k − spheres has been obtained for the nonlinear Schrödingerequation. In particular, in [5] the existence of solutions concentrating on a circle has beenobtained when radial symmetry occurs, as in Theorem 1. Our aim is to extend [7, 5] tothe nonlinear Schrödinger-Poisson system.Now we describe our assumptions.1.1. The potentials.
We consider a nonnegative potential V ∈ C ( R \ { } ) , a non-negative competing function K ∈ C ( R \ { } ) , K , and a weight ρ ∈ L / loc ( R ) ∩ L ∞ loc (cid:0) R \ { } (cid:1) . We assume that for every R ∈ O (3) such that R ( d ) = d , we have V ◦ R = V , K ◦ R = K and ρ ◦ R = ρ . This will be the case if for example V , K and ρ are radial functions.1.2. The nonlinearity.
We consider, for simplicity, a homogeneous nonlinear term u p with < p < ∞ . The condition p > will be needed in order to ensure the boundednessof Palais-Smale sequences.1.3. The growth conditions.
Let W ( x ) := V ( x ) + ρ ( x )1 + | x | . Following [7, 13] we impose one of the three sets of growth conditions at infinity : ( G ∞ ) there exists σ < p − such that lim sup | x |→∞ K ( x ) | x | σ < ∞ ;( G ∞ ) there exists σ ∈ R such that lim inf | x |→∞ W ( x ) | x | > and lim sup | x |→∞ K ( x ) | x | σ < ∞ ;( G ∞ ) there exist α < and σ ∈ R such that lim inf | x |→∞ W ( x ) | x | α > and lim sup | x |→∞ K ( x )exp( σ | x | − α ) < ∞ . Note that in comparison with [7], in ( G ∞ ) and ( G ∞ ) , V might vanish somewhere. Wealso impose one of the three sets of growth conditions at the origin, which mirror those atinfinity : ( G ) there exists τ > − , such that lim sup | x |→ K ( x ) | x | τ < ∞ , ( G ) there exists τ ∈ R such that lim inf | x |→ V ( x ) | x | > and lim sup | x |→ K ( x ) | x | τ < ∞ ;( G ) there exist γ > and τ ∈ R such that lim inf | x |→ V ( x ) | x | γ > and lim sup | x |→ K ( x )exp( τ | x | − γ − ) < ∞ . ONCENTRATION ON CIRCLES FOR NONLINEAR SCHRÖDINGER-POISSON 3
The auxiliary potential.
Before we can state our last assumption, we need a fewpreliminaries. Let a, b > . We consider the limit equation − ∆ u + au = bu p in R . (2)The weak solutions of (2) are critical points of the functional I a,b : H ( R ) → R definedby I a,b ( u ) := 12 Z R (cid:0) |∇ u | + au (cid:1) dx − bp + 1 Z R u p +1 dx. (3)Any nontrivial critical point u ∈ H ( R ) of I a,b , belongs to the Nehari manifold N a,b := (cid:8) u ∈ H ( R ) | u and hI ′ a,b ( u ) , u i = 0 (cid:9) . A solution u ∈ H ( R ) is a least-energy solution of (2) if I a,b ( u ) = inf v ∈N a,b I a,b ( v ) . The ground-energy function is defined by E : R + × R + → R + : ( a, b )
7→ E ( a, b ) := inf u ∈N a,b I a,b ( u ) . It is standard to show that E ( a, b ) = inf γ ∈ Γ a,b max t ∈ [0 , I a,b ( γ ( t )) , (4)where Γ a,b := (cid:8) γ ∈ C ([0 , , H ( R )) | γ (0) = 0 , I a,b ( γ (1)) < (cid:9) . The auxiliary potential M : R → (0 , + ∞ ] is defined by ( x ′ , x ′′ )
7→ M ( x ′ , x ′′ ) := (cid:26) | x ′′ |E ( V ( x ) , K ( x )) if K ( x ) > , + ∞ if K ( x ) = 0 . The following lemma states some properties of the ground-energy function, see [7, Lemma3].
Lemma 1.1.
For every ( a, b ) ∈ R +0 × R +0 , E ( a, b ) is a critical value of I a,b and we have E ( a, b ) = inf u ∈ H ( R ) u =0 max t ≥ I a,b ( tu ) . If u ∈ N a,b and E ( a, b ) = I a,b ( u ) , then u ∈ C ( R ) and up to a translation, u is a radialfunction such that ∇ u ( x ) · x < for every x ∈ R \ { } . Moreover, the following propertieshold: (i) E is continuous in R +0 × R +0 ; (ii) for every b ∗ ∈ R +0 , a → E ( a, b ∗ ) is strictly increasing; (iii) for every a ∗ ∈ R +0 , b → E ( a ∗ , b ) is strictly decreasing; (iv) for every λ > , E ( λa, λb ) = λ − / E ( a, b ) ; (v) the ground-energy function satisfies E ( a, b ) = E (1 , a p +1 p − − b − p − . The last property of the preceding lemma implies the following explicit form of theauxiliary potential: M ( x ′ , x ′′ ) = E (1 , | x ′′ | [ V ( x )] p +1 p − − [ K ( x )] − p − . Due to the symmetry that we shall impose on the solution (see (14)), the concentrationcan only occur in the plane π . We assume that there exists a smooth bounded open set Λ ⊂ R such that ¯Λ ∩ d = ∅ , Λ ∩ π = ∅ , (5)for every R ∈ O (3) such that R ( d ) = d , R (Λ) = Λ (6) DENIS BONHEURE, JONATHAN DI COSMO, AND CARLO MERCURI and the following inequalities hold < inf Λ ∩ π M < inf ∂ Λ ∩ π M , (7) inf Λ ∩ π M < Λ M . (8)By continuity of M in Λ , this last condition is not restrictive. Similarly, we can alsoassume that V > on Λ and that M is continuous on Λ .Our main result is the following. Theorem 2.
Let p > and V, K and ρ be functions satisfying the assumptions in 1.1.Assume that one set ( G i ∞ ) of growth conditions at infinity and one set ( G j ) of growthconditions at the origin hold. Assume also that there exists an open bounded set Λ ⊂ R such that (5) , (6) , (7) and (8) hold. Then there exists ε > such that for every <ε < ε , problem (1) has at least one positive solution u ε . Moreover, for every < ε < ε ,there exists x ε ∈ Λ ∩ π such that u ε attains its maximum at x ε , lim inf ε → u ε ( x ε ) > , lim ε → M ( x ε ) = inf Λ ∩ π M , and there exist C > and λ > such that u ε ( x ) ≤ C exp (cid:18) − λε d ( x, S ε )1 + d ( x, S ε ) (cid:19) (cid:0) | x | (cid:1) − , ∀ x ∈ R , where S ε is the circle centered at the origin, contained in the plane π and of radius | x ′′ ε | . In Section , we deal with an auxiliary penalized problem. This by now classicalpenalization argument goes back to del Pino and Felmer [10]. The method has then beenadapted in [7, 13] in the frame of vanishing or compactly supported potentials. Section is devoted to the asymptotic analysis of the solutions of the penalized problem while inSection , we show how to go back to the original problem. At last, we give some finalcomments in Section . Throughout the paper, we use the following notation :- D , ( R ) is the space { u ∈ L ∗ ( R ) : ∇ u ∈ L ( R ; R ) } equipped with the norm || u || D , ( R ) := ||∇ u || L ( R ) ; - L qQ ( R ) is the Lebesgue space of measurable functions such that Z R Q ( x ) | u | q dx < ∞ . As usual, u + := max( u, and u − := max( − u, , B R is the open ball of radius R and c , c , ...c j , C , C , ...C k always denote positive real constants.2. Existence for the penalized problem
Following [13] and [5], we define the penalization potential H : R → R by H ( x ) := κ | x | (cid:16)(cid:0) log | x | (cid:1) + 1 (cid:17) β where β > and < κ < . Notice that for all x ∈ R , we have H ( x ) ≤ κ | x | . By Hardy’s inequality, we deduce that the quadratic form associated to − ∆ − H is positive,i.e. Z R (cid:0) |∇ u | − Hu (cid:1) ≥ (cid:18) − κ (cid:19) Z R | u ( x ) | | x | dx ≥ , (9)for all u ∈ D , ( R ) .This inequality implies the following comparison principle. ONCENTRATION ON CIRCLES FOR NONLINEAR SCHRÖDINGER-POISSON 5
Proposition 2.1.
Let Ω ⊂ R \ { } be a smooth domain. Let v, w ∈ H loc (Ω) ∩ C (Ω) besuch that ∇ ( w − v ) − ∈ L (Ω) , ( w − v ) − / | x | ∈ L (Ω) and − ∆ w − Hw ≥ − ∆ v − Hv, ∀ x ∈ Ω . (10) If ∂ Ω = ∅ , assume also that w ≥ v on ∂ Ω . Then w ≥ v in Ω .Proof. It suffices to multiply the inequality (10) by ( w − v ) − , integrate by parts and use(9). (cid:3) Fix µ ∈ (0 , . We define the penalized nonlinearity g ε : R × R → R by g ε ( x, s ) := χ Λ ( x ) K ( x ) s p + + (1 − χ Λ ( x )) min (cid:8)(cid:0) ε H ( x ) + µV ( x ) (cid:1) s + , K ( x ) s p + (cid:9) . Let G ε ( x, s ) := R s g ε ( x, σ ) dσ . One can check that g ε is a Carathéodory function with thefollowing properties :( g ) g ε ( x, s ) = o ( s ) , s → + , uniformly in compact subsets of R .( g ) there exists p > such that lim s →∞ g ε ( x, s ) s p = 0 , ( g ) there exists < θ ≤ p + 1 such that < θG ε ( x, s ) ≤ g ε ( x, s ) s ∀ x ∈ Λ , ∀ s > , < G ε ( x, s ) ≤ g ε ( x, s ) s ≤ (cid:0) ε H ( x ) + µV ( x ) (cid:1) s ∀ x / ∈ Λ , ∀ s > , ( g ) the function s g ε ( x, s ) s is nondecreasing for all x ∈ R .Now we use Critical Point Theory in order to find solutions to the penalized problem − ε ∆ u + V ( x ) u + ρ ( x ) φu = g ε ( x, u ) , x ∈ R , − ∆ φ = ρ ( x ) u . (11)For any u ρ ∈ L loc ( R ) such that Z R Z R u ( x ) u ( y ) ρ ( x ) ρ ( y ) | x − y | dx dy < ∞ , the standard distributional solution φ u := 14 π | x | ⋆ u ρ (12)belongs to D , ( R ) and is a weak solution in D , (e.g. [16]). Since we consider u ∈ D , ,we will assume ρ ∈ L / loc . A suitable choice of the space X of admissible functions is givenin the work of Ruiz [16], in the case of V ≡ and ρ ≡ . Inspired by [16], we define, formeasurable
V, ρ ≥ , k u k X := Z R |∇ u | + V ( x ) u dx + (cid:16) Z R Z R u ( x ) u ( y ) ρ ( x ) ρ ( y ) | x − y | dx dy (cid:17) / and X := { u ∈ D , ( R ) : k u k X < ∞} . (13)As pointed out in [16], the space X is a uniformly convex Banach space, hence it isreflexive. Precisely, we look for solutions ( u, φ u ) ∈ E × D , ( R ) . We also define H V,ε to be the closure of D ( R ) with respect to the norm k u k H V,ε := Z R ε |∇ u | + V ( x ) u dx. We will focus on the closed subspace E ⊂ X of functions which are radial in π , namely E := (cid:8) u ∈ X | ∀ R ∈ O (3) s.t. R ( d ) = d, u ◦ R = u (cid:9) . (14) DENIS BONHEURE, JONATHAN DI COSMO, AND CARLO MERCURI
Solutions of (11) are the critical points of the functional J ε ( u ) := 12 Z R ( ε |∇ u | + V ( x ) u ) dx + 14 Z R φ u u ρ ( x ) dx − Z R G ε ( x, u ) dx, which is C ( X ; R ) .In the present section we find critical points for J ε through a minimax scheme used in[6], modeled on [3].The main result of this section is Theorem 3.
Assume
V, K, ρ satisfy the assumptions of Theorem 2. Then, for any ε > and κ, µ > small enough, there exists a critical point for J ε at level c ε := inf γ ∈ Γ max t ∈ [0 , J ε ( γ ( t )) , (15) where Γ ε := { γ ∈ C ([0 , , E ) : γ (0) = 0 , J ε ( γ (1)) < } , (16) corresponding to a nontrivial solution ( u, φ ) ∈ E × D , ( R ) for (11). Moreover u ispositive.Remark . Due to the invariance of the Lebesgue measure by rotations, by the symmetriccriticality principle [14] , if u ∈ E is critical for J ε | E , then u is also critical for J ε | X .The functional J ε has the mountain pass geometry, as it is shown in the following Lemma 2.2.
The functional J ε satisfies the mountain pass geometry for any p > , pro-vided κ, µ > are small enough. Furthermore, there exists a Palais-Smale (P-S) sequenceat the minimax level c ε . In particular, defining S := { u ∈ E : u − ≡ } ,S /n := { u ∈ E : inf y ∈ S k u − y k E < /n } , it is possible to select the P-S sequence ( u n ) n in such a way that u k ∈ S /k for all k ∈ N .Proof. We first prove that, for any p > , the origin is a local minimum for J ε . Notice that k u k p +1 L p +1 K (Λ) ≤ C k u k p +1 E . Furthermore by Lemma 2.3 below, taking κ, µ > small enough,we have J ε ( u ) ≥ c Z R ( ε |∇ u | + V ( x ) u ) dx + 14 Z R φ u u ρ ( x ) dx − C k u k p +1 E . Since, by definition, we have ( R R φ u u ρ ( x ) dx ) / = k u k E − k u k H V,ε , we get J ε ( u ) ≥ c k u k H V,ε + 14 h k u k E − k u k H V,ε i − C k u k p +1 E = c k u k H V,ε + 14 k u k E − k u k H V,ε k u k E + 14 k u k H V,ε − C k u k p +1 E . Therefore, we get J ε ( u ) ≥ c k u k H V,ε − α − k u k H V,ε + α − α k u k E − C k u k p +1 E . Let k u k E < δ. Then we have J ε ( u ) ≥ (cid:20) c − α − δ (cid:21) k u k H V,ε + (cid:20) α − α − Cδ p − (cid:21) k u k E . This yields, for α > and δ small enough, J ε ( u ) ≥ (cid:20) α − α − Cδ p − (cid:21) k u k E . Hence, the origin is a strict local minimum point for J ε . Moreover, J ε attains negative values along curves of the form u t := tu , with u ∈ E such that u + and t > . Hence J ε has the mountain pass geometry. ONCENTRATION ON CIRCLES FOR NONLINEAR SCHRÖDINGER-POISSON 7
By the general minimax principle [18, p.41], there exists a P-S sequence ( u n ) n suchthat, if for γ n ∈ Γ , max t ∈ [0 , J ε ( γ n ( t )) ≤ c ε + 1 n , then dist ( u n , γ n ([0 , < n . (17)Finally, since J ε ( u ) = J ε ( | u | ) , the conclusion follows from (17). (cid:3) Lemma 2.3.
For any positive constants c > there exists κ ( c ) such that c Z R |∇ u | dx ≥ Z R \ Λ H ( x ) u dx, ∀ κ < κ ( c ) , ∀ u ∈ D , ( R ) , Proof.
The claim follows directly from Hardy’s inequality. (cid:3)
We now study some properties of the P-S sequences found in Lemma 2.2.
Lemma 2.4.
Let ( u n ) n be as in Lemma 2.2 such that u k ∈ S /k for all k ∈ N . Then ( a ) Z R φ ( u n ) − ( x )( u n ) − ρ ( x ) dx → . Furthermore, ( u n ) n is bounded in E, provided κ, µ > are small enough, and we have ( b ) Z R φ u n ( x )( u n ) − ρ ( x ) dx → . Proof.
By definition, there exists a sequence ( y n ) n ⊂ S such that k u n − y n k E → . Hence ( a ) follows: Z R φ ( u n ) − ( x )( u n ) − ρ ( x ) dx = Z R Z R ( u n ) − ( x ) ρ ( x )( u n ) − ( z ) ρ ( z ) | x − z | dxdz ≤ Z R Z R ( u n − y n ) ( x ) ρ ( x )( u n − y n ) ( z ) ρ ( z ) | x − z | dxdz ≤ k u n − y n k E → Notice that ( b ) follows if we prove that ( u n ) n is bounded. Indeed, define, for f, g measur-able and nonnegative functions, the following quantity D ( f, g ) := Z R Z R f ( x ) | x − y | − g ( y ) dxdy. From [11, p.250], we have | D ( f, g ) | ≤ D ( f, f ) D ( g, g ) . (18)If ( u n ) n is bounded in E, by the inequality above with f := u n ρ and g := ( u n ) − ρ andby ( a ) we have Z R φ u n ( x )( u n ) − ρ ( x ) dx ≤ C Z R φ ( u n ) − ( x )( u n ) − ρ ( x ) dx → We now prove that ( u n ) n is bounded.Define ∆ n := Z R ( g ε ( x, u n ) u n − ( p + 1) G ε ( x, u n )) dx Using Lemma 2.3, we have, choosing κ, µ > small enough, ∆ n ≥ − ( p + 1) Z R \ Λ G ε ( x, u n ) dx ≥ − p + 12 Z R \ Λ [ ε H ( x ) + µV ( x )] u n dx ≥ − p − k u n k H V,ε . DENIS BONHEURE, JONATHAN DI COSMO, AND CARLO MERCURI
Since ( u n ) n is a P-S sequence, the above estimate yields C ≥ ( p + 1) J ε ( u n ) − ( J ′ ε ( u n ) , u n )= p − k u n k H V,ε + p − Z R φ u n ( x ) u n ρ ( x ) dx + ∆ n ≥ p − k u n k H V,ε + p − Z R φ u n ( x ) u n ρ ( x ) dx. (19)As a consequence, the claim follows. (cid:3) In the following we shall need a family of cut-off functions. Consider a smooth function ζ ( r ) such that ζ ( r ) = 1 on [2 , ∞ ) and ζ ( r ) = 0 on [0 , . Then define η R ( x ) := ζ (cid:16) log(1 + | x | ) R (cid:17) . One has k| x | · |∇ η R ( x ) |k ∞ ≤ CR . (20)
Lemma 2.5.
Let ( u n ) n ⊂ E be as in Lemma 2.2 and u n ⇀ u ≥ in E . Then, for all δ > , there exists a ball B ⊂ R such that, for all κ, µ > small enough, a ) lim sup n →∞ Z R \ B φ u n ( x ) u n ρ ( x ) dx < δ,b ) lim sup n →∞ Z R \ B V ( x ) u n dx < δ,c ) lim sup n →∞ Z R \ B H ( x ) u n dx < δ,d ) lim sup n →∞ Z R \ B φ u n ( x )( u n ) − u ρ ( x ) dx < δ,e ) lim sup n →∞ Z R \ B φ u n ( x )( u n ) + u ρ ( x ) dx < δ. Proof.
Consider the above family of cut-off functions. We claim that, uniformly in n , Z R (cid:16) |∇ u n | − H ( x ) u n (cid:17) η R dx ≥ O (cid:16) R (cid:17) , R → ∞ , (21)for all κ > small enough.In order to prove this, compute |∇ ( u n η R ) | = η R |∇ u n | + 2 u n η R ∇ u n ∇ η R + u n |∇ η R | . We have (cid:12)(cid:12)(cid:12) Z R u n η R ∇ u n ∇ η R dx (cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12)(cid:12) Z R u n | x | ∇ u n | x |∇ η R dx (cid:12)(cid:12)(cid:12) ≤ k| x |∇ η R k ∞ (cid:12)(cid:12)(cid:12) Z R u n | x | ∇ u n dx (cid:12)(cid:12)(cid:12) . By (20), Cauchy-Schwarz and Hardy inequalities, we obtain (cid:12)(cid:12)(cid:12) Z R u n ∇ u n ∇ η R dx (cid:12)(cid:12)(cid:12) ≤ C R k∇ u n k ≤ C R . (22)Here we take into account that, since u n ⇀ u in E, k∇ u n k is bounded. In the same way,one can easily obtain Z R u n |∇ η R | ≤ C R . Hence, by Hardy’s inequality and the above estimates, we have, as R → ∞ , Z R (cid:16) |∇ u n | − H ( x ) u n (cid:17) η R dx ≥ Z R (cid:16) |∇ ( u n η R ) | − κ ( u n η R ) | x | (cid:17) dx + O (cid:16) R (cid:17) ≥ O (cid:16) R (cid:17) , and the claim follows. Furthermore, simply notice that ONCENTRATION ON CIRCLES FOR NONLINEAR SCHRÖDINGER-POISSON 9 Z R ∇ u n ∇ ( u n η R ) dx = Z R |∇ u n | η R dx + O (cid:16) R (cid:17) , R → ∞ , (23)Finally, by (21) and (23), we have, for µ, κ > small enough, o (1) = ( J ′ ε ( u n ) , u n η R ) ≥ ε Z R |∇ u n | η R dx + ε Z R (cid:16) |∇ u n | − H ( x ) u n (cid:17) η R dx + Z R φ u n u n η R ρ ( x ) dx + (1 − µ ) Z R V ( x ) u n η R dx + O (cid:16) R (cid:17) ≥ ε Z R |∇ u n | η R dx + Z R φ u n u n η R ρ ( x ) dx + (1 − µ ) Z R V ( x ) u n η R dx + O (cid:16) R (cid:17) , as R → ∞ . Hence, taking B := { x ∈ R : | x | ≤ e R } , since all the terms are nonnegative,the above estimates yield statements ( a ) , ( b ) and, using (21), statement ( c ) . In order to prove ( d ) we use Cauchy-Schwarz inequality and (18), obtaining Z R φ u n ( x )( u n ) − u ρ ( x ) η R dx ≤ ( D ( u n , ( u n ) − )) / (cid:16) D ( u n , u n ) D ( u , u ) (cid:17) / → , since D ( u n , u n ) is bounded and D ( u n , ( u n ) − ) → by Lemma 2.4.Finally we prove ( e ) . We have, for any
R > ,o (1) = ( J ′ ε ( u n ) , uη R ) ≥ < u n , uη R > H V,ε + Z R φ u n ( x )( u n ) + u ρ ( x ) η R dx − Z R φ u n ( x )( u n ) − u ρ ( x ) η R dx − Z R ( ε H ( x ) + µV ( x ))( u n ) + uη R dx, n → ∞ . Notice that, by weak convergence, we have, for any R , < u n , uη R > H V,ε → < u, uη R > H V,ε , and Z R ( ε H ( x ) + µV ( x ))( u n ) + uη R dx → Z R ( ε H ( x ) + µV ( x )) u η R dx. Now fix α > small and take R α such that for all R > R α we have Z R ( ε H ( x ) + µV ( x )) u η R dx ≤ α and, by ( d ) , Z R φ u n ( x )( u n ) − u ρ ( x ) η R dx ≤ α. Arguing as for the estimate (22), we can choose R α large enough such that (cid:12)(cid:12)(cid:12) Z R u ∇ u ∇ η R dx (cid:12)(cid:12)(cid:12) ≤ CR < α, for every
R > R α . Hence, writing ∇ ( η R u ) = η R ∇ u + u ∇ η R , the term < u, uη R > H V,ε isthe sum of a positive term plus a small term. Therefore, we obtain o (1) + 3 α ≥ Z R φ u n ( x )( u n ) + u ρ ( x ) η R dx and claim ( e ) follows. This concludes the proof. (cid:3) Arguing as in the above lemmas we have
Lemma 2.6.
Under the assumptions on ρ, V, K given in Theorem 2, let ( u n ) be as in theabove lemma. Then for all δ > , there exists a ball B (0) ⊂ R , such that ( a ) lim sup n →∞ Z B (0) φ u n ( x ) u n ρ ( x ) dx < δ, ( b ) lim sup n →∞ (cid:12)(cid:12)(cid:12) Z B (0) φ u n ( x )( u n ) − uρ ( x ) dx (cid:12)(cid:12)(cid:12) < δ, ( c ) lim sup n →∞ (cid:12)(cid:12)(cid:12) Z B (0) φ u n ( x )( u n ) + uρ ( x ) dx (cid:12)(cid:12)(cid:12) < δ. Lemma 2.7.
Let ( u n ) n be as in Lemma 2.5. Then, passing if necessary to a subsequence,we have k u n k H V,ε → k u k H V,ε . Proof.
Since u n ⇀ u in H V,ε , for some subsequence, we have o (1) = ( J ′ ε ( u n ) , u n − u ) = k u n k H V,ε − k u k H V,ε + o (1)+ Z R φ u n ( x ) u n ( u n − u ) ρ ( x ) dx + Z R g ε ( x, u n )( u n − u ) dx. (24)We show that A n := Z R g ε ( x, u n )( u n − u ) dx → and B n := Z R φ u n ( x ) u n ( u n − u ) ρ ( x ) dx → . Observe that A n := Z Λ ... + Z B \ Λ ... + Z R \ B ..., for some large ball B containing Λ . Since ( u n ) n is bounded in E and E is compactlyembedded in L q (Λ) for all q > , passing to a subsequence, we can assume u n → u in L p +1 (Λ) . As a consequence, passing if necessary to a subsequence, we have | g ε ( x, u n ) | E ֒ → L ε H + µV ( B \ Λ) , it follows that Z B \ Λ g ε ( x, u n )( u n − u ) dx → . Finally, taking B large and using ( b ) , ( c ) in Lemma 2.5, we have Z R \ B g ε ( x, u n )( u n − u ) dx → , hence A n → . In order to prove B n → , we use a similar splitting argument. Fix δ > .B n = Z B (0) ... + Z B \ B (0) ... + Z R \ B ... = I ,n + I ,n + I ,n , Now we choose B such that, using Lemma 2.5, we have | I ,n | < δ. Shrinking the ball B (0) if necessary, we infer from Lemma 2.6 that | I ,n | < δ .Next, we estimate I ,n as follows. By Hölder and Sobolev inequalities, we have Z B \ B (0) φ u n ( x ) | u n ( u n − u ) | ρ ( x ) dx ≤ C k ρ k L ∞ ( B \ B (0)) k φ u n k D , ( R ) k u n ( u n − u ) k L / ( B \ B (0)) . (25) ONCENTRATION ON CIRCLES FOR NONLINEAR SCHRÖDINGER-POISSON 11 Due to the weak convergence in E , k φ u n k D , ( R ) is bounded, and therefore, by compact-ness, we get | I ,n | < δ. This concludes the proof. (cid:3) Proof of Theorem 3. By Lemma 2.2 and Lemma 2.4, there exists a bounded P-S sequence ( u n ) n at the minimax level c ε , such that u n ⇀ u in E. We are going to prove that, passingif necessary to a subsequence, i ) J ε ( u n ) → J ε ( u ) ,ii ) J ′ ε ( u ) = 0 . This will imply the existence of a nontrivial solution u. In order to prove i ) , notice that, by Lemma 2.7, it is enough to show that Z R G ε ( x, u n ) dx → Z R N G ε ( x, u ) dx and Z R φ u n ( x ) u n ρ ( x ) dx → Z R φ u ( x ) u ρ ( x ) dx. In order to prove the former limits, we can argue as for the terms involving g ε in Lemma2.7, splitting the integral Z R | G ε ( x, u n ) − G ε ( x, u ) | dx = Z Λ ... + Z B \ Λ ... + Z R \ B ... . We can assume u n → u in L p +1 (Λ) and almost everywhere. Fix δ > . By using thecompact embedding E ֒ → L q (Λ) which holds for all q > , the dominated convergencetheorem yields Z Λ | G ε ( x, u n ) − G ε ( x, u ) | dx < δ for large n and, in the same fashion, using property ( g ) and the compact embedding of E in L ε H + µV ( B \ Λ) , it follows that Z B \ Λ | G ε ( x, u n ) − G ε ( x, u ) | dx < δ for some subsequence, taking n larger if necessary. Finally, observe that, by ( b ) , ( c ) ofLemma 2.5, there exists B large enough, such that for n large enough, Z R \ B | G ε ( x, u n ) − G ε ( x, u ) | dx < δ. Now we prove the second limit. We compute (cid:12)(cid:12)(cid:12)(cid:12)Z R [ φ u n ( x ) u n ρ ( x ) − φ u ( x ) u ρ ( x )] dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) Z B (0) ... (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Z B \ B (0) ... (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Z R \ B ... (cid:12)(cid:12)(cid:12) = J ,n + J ,n + J ,n . Fix δ > . Arguing as in the proof of Lemma 2.7, we can take B large enough in such away that Z R \ B φ u ( x ) u ρ ( x ) dx < δ, yielding, with Lemma 2.5, J ,n ≤ (cid:12)(cid:12)(cid:12) Z R \ B φ u n ( x ) u n ρ ( x ) dx (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Z R \ B φ u ( x ) u ρ ( x ) dx (cid:12)(cid:12)(cid:12) < δ. Since we can shrink B (0) so that Z B (0) φ u ( x ) u ρ ( x ) dx < δ, we deduce from Lemma 2.6 that J ,n ≤ (cid:12)(cid:12)(cid:12) Z B (0) φ u n ( x ) u n ρ ( x ) dx (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Z B (0) φ u ( x ) u ρ ( x ) dx (cid:12)(cid:12)(cid:12) < δ. Now, with by now familiar arguments, we can estimate J ,n . Indeed, using Hölder andSobolev inequalities, we have Z B \ B (0) | φ u n ( x ) u n − φ u ( x ) u | ρ ( x ) dx ≤ Z B \ B (0) | φ u n ( x ) u n − φ u n ( x ) u | ρ ( x ) dx + Z B \ B (0) | φ u n ( x ) u − φ u ( x ) u | ρ ( x ) dx ≤ C k ρ k L ∞ ( B \ B (0)) k φ u n k D , ( R ) k u n − u k L / ( B \ B (0)) ++ k ρ k L ∞ ( B \ B (0)) Z B \ B (0) | φ u n ( x ) u − φ u ( x ) u | dx. (26)By compactness, we infer that k u n − u k L / ( B \ B (0)) → , while ( φ u n ) n is bounded in D , ( R ) , hence the first term in (26) goes to zero. On the other hand, since φ u n ⇀ φ u in D , ( R ) , we have φ u n → φ u strongly in L d ( B \ B (0)) for any d < ∗ . Hence, Hölderinequality implies the last term in (26) goes also to zero. As a consequence J ,n ≤ δ andthis yields i ) . The proof of ii ) is rather standard, using the weak convergence in E and the samesplitting arguments. The maximum principle implies u > on R . This completes theproof of the theorem. (cid:3) Asymptotics of solutions In order to show that the solution u ε found in Theorem 3 satisfies, for ε small enough,the original problem and concentrates around a circle, we need to study the asymptoticbehaviour of u ε as ε → . Since many arguments are similar to the ones in [5], we onlystress the differences with these. We begin with an energy estimate. Proposition 3.1 (Upper estimate of the critical value) . Suppose that the assumptions ofTheorem 3 are satisfied. For ε small enough, the critical value c ε defined in (15) satisfies c ε ≤ ε (cid:0) π inf Λ ∩ π M + o (1) (cid:1) as ε → . Moreover, the solution u ε of (11) found in Theorem satisfies, for some C > , k u ε k H V,ε ≤ Cε (27) and Z R |∇ φ u ε | dx = Z R Z R u ε ( x ) u ε ( y ) ρ ( x ) ρ ( y ) | x − y | dx dy ≤ Cε . (28) Proof. Take x = (0 , x ′′ ) ∈ Λ ∩ π such that M ( x ) = inf Λ ∩ π M . Denote by I thefunctional defined by (3) with a = V ( x ) and b = K ( x ) and let c := E ( V ( x ) , K ( x )) .From (4), we infer that for every δ > , there exists a continuous path γ δ : [0 , → H ( R ) such that γ δ (0) = 0 , I ( γ δ (1)) < and c ≤ max t ∈ [0 , I ( γ δ ( t )) ≤ c + δ. Let η ∈ D (cid:0) R (cid:1) be a cut-off function with support in Λ such that ≤ η ≤ , η = 1 in aneighbourhood of (0 , | x ′′ | ) and k∇ η k ∞ ≤ C . We consider the path ¯ γ δ ( t ) : x → η ( x ′ , | x ′′ | ) γ δ ( t ) (cid:18) x ′ ε , | x ′′ | − | x ′′ | ε (cid:19) . Setting ¯ γ δ ( t )( x ′ , x ′′ ) =: v t (cid:18) x ′ ε , | x ′′ | − | x ′′ | ε (cid:19) , ONCENTRATION ON CIRCLES FOR NONLINEAR SCHRÖDINGER-POISSON 13 we compute, by a change of variable, Z R ( ε |∇ ¯ γ δ ( t ) | + V ( x )¯ γ δ ( t ) ) dx − Z R G ε ( x, ¯ γ δ ( t )) dx = π Z R Z ∞− | x ′′ | ε (cid:0) |∇ v t | + V ( εy ′ , εσ + | x ′′ | ) v t (cid:1) ( εσ + | x ′′ | ) ε dσ εdy ′ − π Z R Z ∞− | x ′′ | ε G ( εy ′ , εσ + | x ′′ | , v t )( εσ + | x ′′ | ) ε dσ εdy ′ . The boundedness of ρ in Λ and Hardy-Littlewood-Sobolev inequality leads to Z R Z R ¯ γ δ ( t )( x ) ¯ γ δ ( t )( y ) ρ ( x ) ρ ( y ) | x − y | dx dy ≤ k ρ k L ∞ (Λ) Z R Z R v t ( x ′ , σ ) v t ( y ′ , τ ) ε ( | x ′ − y ′ | + | σ − τ | ) ǫ dx ′ dy ′ dσ dτ ≤ Cε k v t k L ( R ) = o (cid:0) ε (cid:1) . For ε small enough, we obtain ε − J ε (¯ γ δ ( t )) ≤ ω k | x ′′ | k I ( γ δ ( t )) + o (1) . (29)It follows that for ε small enough, ¯ γ δ belongs to the class of paths Γ ε defined by (16). Wededuce from (15) that ε − c ε ≤ max t ∈ [0 , ε − J ε (¯ γ δ ( t )) ≤ π | x ′′ | max t ∈ [0 , I ( γ δ ( t )) + o (1) ≤ π | x ′′ | ( c + δ ) + o (1) . Since δ > is arbitrary and | x ′′ | c = M ( x ) , the first statement is established. Thesecond statement is proved by a computation similar to (19). (cid:3) Proposition 3.2 (No uniform convergence to in Λ ) . Suppose that the assumptions ofTheorem are satisfied and let ( u ε ) ε ⊂ E be positive solutions of (11) obtained in Theorem . Then there exists δ > such that k u ε k L ∞ (Λ) ≥ δ. Proof. See [5, Proposition 4.2]. (cid:3) By the symmetry imposed on the solution u ε , one can write u ε ( x ′ , x ′′ ) = ˜ u ε ( x ′ , | x ′′ | ) with ˜ u ε : R → R . Notice that φ u ε has the same symmetry as u ε , i.e. for all R ∈ O (3) such that R ( d ) = d , φ u ε ◦ R = φ u ε . This follows easily from the representation formula(12).Since the H V,ε -norm of u ε is of the order ε , it is natural to rescale ˜ u ε ( x ′ , | x ′′ | ) as ˜ u ε ( x ′ ε + εy ′ , | x ′′ ε | + ε | y ′′ | ) around a well-chosen family of points x ε = ( x ′ ε , x ′′ ε ) ∈ R . Thenext lemma shows that the sequences of rescaled solutions converge, up to a subsequence,in C ( R ) to a function v ∈ H ( R ) . Lemma 3.3. Suppose that the assumptions of Theorem 3 are satisfied. Let ( u ε ) ε ⊂ E bepositive solutions of (11) found in Theorem 3, ( ε n ) n ⊂ R + and ( x n ) n ⊂ R be sequencessuch that ε n → and x n = ( x ′ n , x ′′ n ) → ¯ x = (¯ x ′ , ¯ x ′′ ) ∈ ¯Λ as n → ∞ . Set Ω n := R × i − | x ′′ n | ε n , + ∞ h and let v n : Ω n → R be defined by v n ( y, z ) := ˜ u ε n ( x ′ n + ε n y, | x ′′ n | + ε n z ) , (30) where ˜ u ε n : R → R is such that u ε n ( x ′ , x ′′ ) = ˜ u ε n ( x ′ , | x ′′ | ) . Then, there exists v ∈ H ( R ) such that, along a subsequence that we still denote by ( v n ) n , v n C ( R ) −→ v. Moreover, v is a solution of the equation − ∆ v + V (¯ x ) v = K (¯ x ) v p , ¯ x ∈ R . Proof. We infer from Proposition 3.1 that for all n ∈ N , Z Ω n (cid:0) |∇ v n ( y, z ) | + V ( x ′ n + ε n y, | x ′′ n | + ε n z ) | v n ( y, z ) | (cid:1) dy dz ≤ C, (31)with C > independent of n .Observe also that each v n solves the equation − ∆ v n − ε n z ∂v n ∂z + V ( x ′ n + ε n y, | x ′′ n | + ε n z ) v n + ρ ( x ′ n + ε n y, | x ′′ n | + ε n z ) ˜ φ n v n = g ε n ( x ′ n + ε n y, | x ′′ n | + ε n z, v n ) , x ∈ R , (32)where we have set ˜ φ n ( y, z ) = φ u εn ( x ′ n + ε n y, | x ′′ n | + ε n z ) . As a consequence of (28), thesequence ( ˜ φ n ) n converges to zero in D , ( R ) . It follows then from Hölder inequality thatthe term ( ρ ( x ′ n + ε n y, | x ′′ n | + ε n z ) ˜ φ n v n ) n is bounded in L loc (cid:0) R \ { } (cid:1) .Define a cut-off function η R ∈ D ( R ) such that ≤ η R ≤ , η R ( x ) = 1 if | x | ≤ R/ , η R ( x ) = 0 if | x | ≥ R and k∇ η R k ∞ ≤ C/R for some C > . Choose ( R n ) n such that R n → ∞ and ε n R n → . Since ¯ x ∈ Λ and ¯Λ ∩ d = ∅ , one has ε n R n ≤ | x ′′ n | if n is largeenough. Define w n ∈ H ( R ) by w n ( y ) := η R n ( y ) v n ( y ) . It was shown in [5, Lemma 4.3] that (31) implies that ( w n ) n is bounded in H ( R ) .Since w n solves equation (32) on B (0 , R n ) for all n , classical regularity estimates yieldthat for every R > and every q > , sup n ∈ N k v n k W ,q ( B (0 ,R )) < ∞ . (33)Up to a subsequence, we can now assume that ( w n ) n converges weakly in H ( R ) tosome function v ∈ H ( R ) . By (33), for every compact set K ⊂ R , w n converges to v in C ( K ) . Moreover, for n large enough, w n = v n in K so that v n → v in C ( K ) . (cid:3) For x, y ∈ R , denote by d d ( x, y ) := q | x ′ − y ′ | + ( | x ′′ | − | y ′′ | ) . the distance between the circles centered at x ′ and y ′ , and of radius | x ′′ | and | y ′′ | respec-tively. We denote by B d the balls for the distance d d , i.e., B d ( x, r ) = { y ∈ R : d d ( x, y ) < r } . We are now going to estimate from below the critical value c ε . In the next two lemmaswe estimate the action respectively inside and outside neighbourhoods of points. Lemma 3.4. Suppose that the assumptions of Theorem 3 are satisfied. Let u ε ∈ E bepositive solutions of (11) found in Theorem 3, ( ε n ) n ⊂ R + and ( x n ) n ⊂ R n be sequencessuch that ε n → and x n = ( x ′ n , x ′′ n ) → ¯ x = (¯ x ′ , ¯ x ′′ ) ∈ ¯Λ as n → ∞ . If lim inf n →∞ u ε n ( x n ) > , (34) then we have, up to a subsequence, lim inf R →∞ lim inf n →∞ ε − n Z T n ( R ) (cid:0) ε n |∇ u ε n | + V u ε n (cid:1) − G ε n ( x, u ε n ) ! ≥ π M (¯ x ) , where T n ( R ) := B d ( x n , ε n R ) .Proof. The proof is the same as the one of Lemma . in [5]. Indeed, the Poisson term ispositive and the equation satisfied by the limit of the sequence of rescaled solutions is thesame as in [5]. (cid:3) ONCENTRATION ON CIRCLES FOR NONLINEAR SCHRÖDINGER-POISSON 15 Lemma 3.5. Suppose that the assumptions of Theorem 3 are satisfied. Let u ε ∈ E bepositive solutions of (11) found in Theorem 3, ( ε n ) n ⊂ R + and ( x in ) n ⊂ R be sequencessuch that ε n → and for ≤ i ≤ M , x in → ¯ x i ∈ ¯Λ as n → ∞ . Then, up to a subsequence,we have lim inf R →∞ lim inf n →∞ ε − n Z R \T n ( R ) (cid:0) ε n |∇ u ε n | + V u ε n (cid:1) − G ε n ( x, u ε n ) ! ≥ , where T n ( R ) := S Ki =1 B d ( x in , ε n R ) .Proof. Since the Poisson term is positive, the proof is the same as the one of Lemma . in [5]. (cid:3) Proposition 3.6 (Lower estimate of the critical value) . Suppose that the assumptionsof Theorem 3 are satisfied. Let u ε ∈ E be positive solutions of (11) found in Theorem3, ( ε n ) n ⊂ R + and ( x in ) n ⊂ R be sequences such that ε n → and for ≤ i ≤ M , x in → ¯ x i ∈ ¯Λ as n → ∞ . If for every ≤ i < j ≤ M , we have lim sup n →∞ d d ( x in , x jn ) ε n = ∞ and if for every ≤ i ≤ M , lim inf n →∞ u ε n ( x in ) > , then the critical value c ε defined in (15) satisfies lim inf n →∞ ε − n c ε n ≥ π M X i =1 M (¯ x i ) . Proof. This is a consequence of the two previous lemmas and of the positivity of thePoisson term, see [7, Proposition 16] for the details. (cid:3) Now we can state a first concentration result. It will be completed in the next sectionby a decay estimate. Proposition 3.7 (Uniform convergence to outside small balls) . Suppose that the as-sumptions of Theorem 3 are satisfied and that Λ satisfies the assumptions (5) - (8) . Let ( u ε ) ε ⊂ E be positive solutions of (11) obtained in Theorem 3. If ( x ε ) ε> ⊂ Λ is such that lim inf ε → u ε ( x ε ) > , then(i) lim ε → M ( x ε ) = inf Λ ∩ π M ,(ii) lim ε → x ε ,π ) ε = 0 ,(iii) lim inf ε → d d ( x ε , ∂ Λ) > ,(iv) for every δ > , there exists ε > and R > such that, for every ε ∈ (0 , ε ) , k u ε k L ∞ (Λ \ B d ( x ε ,εR )) ≤ δ. Proof. All the assertions are proved by energy comparisons, using only Propositions 3.1and 3.6. A detailed proof can be found in [5, Proposition 4.7]. (cid:3) Solution of the initial problem Linear inequation outside small balls. In this section we prove that for ε smallenough, the solutions of the penalized problem (11) are also solutions of the initial problem(1). We follow the arguments of [13] and [5]. First we notice that the solutions of (11)satisfy a linear inequation outside small balls. As observed in [6, Theorem 5], the function φ u ε satisfies the estimate φ u ε ( x ) ≥ C ε C ′ ε + | x | , for some constants C ε , C ′ ε > . Set W ε ( x ) := (1 − µ ) V ( x ) + C ε ρ ( x ) C ′ ε + | x | . Lemma 4.1. Suppose that the assumptions of Proposition 3.7 are satisfied and let ( u ε ) ε ⊂ E be positive solutions of (11) found in Theorem 3 and ( x ε ) ε> ⊂ Λ be such that lim inf ε → u ε ( x ε ) > . Then there exist R > and ε > such that for all ε ∈ (0 , ε ) , − ε (∆ u ε + Hu ε ) + W ε u ε ≤ in R \ B d ( x ε , εR ) . (35) Proof. Set η := inf x ∈ Λ (cid:18) µV ( x ) K ( x ) (cid:19) p − . Since V and K are bounded positive continuous functions on ¯Λ , η > . By Proposition3.7, we can find ε > and R > such that for all ε ∈ (0 , ε ] , one has u ε ( x ) ≤ η for all x ∈ Λ \ B d ( x ε , εR ) . We conclude that − ε ∆ u ε + (1 − µ ) V u ε + ρφ u ε u ε ≤ − ε ∆ u ε + V u ε + ρφ u ε u ε − Ku pε = 0 in Λ \ B d ( x ε , εR ) . The fact that u ε satisfies (35) in R \ Λ follows directly from thedefinition of the penalized nonlinearity. (cid:3) This lemma suggests that we can compare the solution u ε with supersolutions of theoperator − ε (∆ + H ) + W ε in order to obtain decay estimates of u ε .4.2. Comparison functions. In this section we recall results from [5] about the com-parison functions. The next lemma provides a minimal positive solution of the operator − ∆ − H in R \ ¯Λ . Lemma 4.2. For every ε > , there exists Ψ ε ∈ C (cid:0) ( R \ { } ) \ Λ (cid:1) such that ( − ε (∆Ψ ε + H Ψ ε ) + W ε Ψ ε = 0 in R \ ¯Λ , Ψ ε = 1 on ∂ Λ , and Z R \ Λ (cid:18) |∇ Ψ ε ( x ) | + | Ψ ε ( x ) | | x | (cid:19) dx < ∞ . (36) Moreover, there exists C > such that, for every x ∈ R \ Λ and every ε > , < Ψ ε ( x ) ≤ C | x | . (37) Proof. See [5, Lemma 5.2]. (cid:3) As explained in [13], the estimate (37) is the best one can hope for, at least if W ε decaysrapidly at infinity. However, if W ε decays quadratically or subquadratically at infinity, wecan improve (37). Lemma 4.3. Let Ψ ε be given by Lemma 4.2.(1) If lim inf | x |→∞ W ε ( x ) | x | > , then there exist λ > , R > and C > such that forevery ε > and x ∈ R \ B (0 , R ) , Ψ ε ( x ) ≤ C (cid:18) R | x | (cid:19) + r − κ + λ ε . (2) If lim inf | x |→∞ W ε ( x ) | x | α > with α < , then there exist λ > , R > , C > and ε > such that for every ε ∈ (0 , ε ) and x ∈ R \ B (0 , R ) , Ψ ε ( x ) ≤ C exp (cid:18) − λε (cid:16) | x | − α − R − α (cid:17)(cid:19) . ONCENTRATION ON CIRCLES FOR NONLINEAR SCHRÖDINGER-POISSON 17 (3) If lim inf | x |→ W ε ( x ) | x | > , then there exist λ > , r > and C > such that forevery ε > and x ∈ B (0 , r ) , Ψ ε ( x ) ≤ C (cid:18) | x | r (cid:19) r − κ + λ ε − . (4) If lim inf | x |→ W ε ( x ) | x | α > with α < , then there exist λ > , R > , C > and ε > such that for every ε ∈ (0 , ε ) and x ∈ B (0 , r ) , Ψ ε ( x ) ≤ C exp (cid:18) − λε (cid:16) | x | − α − − r − α − (cid:17)(cid:19) . Proof. See [5, Lemma 5.3]. (cid:3) Now we provide a comparison function that describes the exponential decay of u ε inside Λ . Lemma 4.4. Let ¯ x ∈ Λ and R > be such that B d (¯ x, R ) ⊂ Λ . (38) Define Φ ¯ xε ( x ) := cosh (cid:18) λ R − d d ( x, ¯ x ) ε (cid:19) . (39) There exists λ > and ε > such that for every ε ∈ (0 , ε ) , one has − ε ∆Φ ¯ xε + W ε Φ ¯ xε ≥ in B d (¯ x, R ) . Proof. See [5, Lemma 5.4]. (cid:3) Lemma 4.5. Let ( x ε ) ε ⊂ Λ be such that lim inf ε → d d ( x ε , ∂ Λ) > and R > . Then, there exist ε > and a family of functions ( w ε ) <ε<ε ⊂ C , loc (( R \{ } ) \ B d ( x ε , εR )) such that for all ε ∈ (0 , ε ) , one has(i) w ε satisfies the inequation − ε (∆ + H ) w ε + W ε w ε ≥ in R \ B d ( x ε , εR ) , (ii) ∇ w ε ∈ L ( R \ B d ( x ε , εR )) and w ε | x | ∈ L ( R \ B d ( x ε , εR )) ,(iii) w ε ≥ on ∂B d ( x ε , εR ) ,(iv) for every x ∈ B d ( x ε , εR ) , w ε ( x ) ≤ C exp (cid:18) − λε d d ( x, x ε )1 + d d ( x, x ε ) (cid:19) (1 + | x | ) − , x ∈ R . Moreover,(1) If lim inf | x |→∞ W ε ( x ) | x | > , then there exists λ > , ν > and C > such that for ε > small enough, w ε ( x ) ≤ C exp (cid:18) − λε d d ( x, x ε )1 + d d ( x, x ε ) (cid:19) (1 + | x | ) − νε . (2) If lim inf | x |→∞ W ε ( x ) | x | α > with α > , then there exists λ > and C > suchthat for ε > small enough, w ε ( x ) ≤ C exp (cid:18) − λε d d ( x, x ε )1 + d d ( x, x ε ) (1 + | x | ) − α (cid:19) . (3) If lim inf | x |→ W ε ( x ) | x | > , then there exists λ > , ν > and C > such that for ε > small enough, w ε ( x ) ≤ C exp (cid:18) − λε d d ( x, x ε )1 + d d ( x, x ε ) (cid:19) (cid:18) | x | | x | (cid:19) νε . (4) If lim inf | x |→ W ε ( x ) | x | α > with α > , then there exists λ > and C > such thatfor ε > small enough, w ε ( x ) ≤ C exp − λε d d ( x, x ε )1 + d d ( x, x ε ) (cid:18) | x | | x | (cid:19) α − ! . Proof. See [5, Lemma 5.5]. (cid:3) Thanks to the previous lemma, we obtain an upper bound on the solutions ( u ε ) ε> of(11). Proposition 4.6. Suppose that the assumptions of Proposition 3.7 are satisfied. Let ( u ε ) ε> ⊂ E be the positive solutions of (11) found in Theorem 3 and ( x ε ) ε> ⊂ Λ be suchthat lim inf ε → u ε ( x ε ) > . Then there exist C > , λ > and ε > such that for all ε ∈ (0 , ε ) , u ε ( x ) ≤ C exp (cid:18) − λε d ( x, S ε )1 + d ( x, S ε ) (cid:19) (1 + | x | ) − , x ∈ R . (40) Moreover, (1) , (2) , (3) and (4) in Lemma 4.5 hold with u ε in place of w ε .Proof. See [5, Lemma 5.6]. (cid:3) Solution of the original problem.Proposition 4.7. Let ( u ε ) ε> ⊂ E be the positive solutions of (11) found in Theorem3. Assume that one set ( G i ∞ ) of growth conditions at infinity and one set ( G j ) of growthconditions at the origin hold. Then there exists ε > such that for all ε ∈ (0 , ε ) , u ε solves the original problem (1) .Proof. We infer from Lemma 3.2 that there exists a family of points ( x ε ) ε> ⊂ Λ suchthat lim inf ε → u ε ( x ε ) > . Assume that the assumptions ( G ∞ ) and ( G ) are satisfied. By Proposition 4.6, we obtainfor ε > small enough and x ∈ R \ Λ , K ( x ) u p − ε ≤ M (1 + | x | ) σ (cid:18) Ce − λε | x | (cid:19) p − ≤ Ce − λε ( p − (1 + | x | ) − ( p − σ ≤ ε κ | x | ((log | x | ) + 1) β = ε H ( x ) . We conclude by definition of the penalized nonlinearity g ε that g ε ( x, u ε ( x )) = K ( x ) u pε ( x ) ,and hence u ε solves the original problem (1). The other cases can be treated in a similarway. (cid:3) Proof of Theorem 2. We proved in Theorem 3 that, for any ε > , the penalized problem(11) possesses a solution u ε . By Proposition 4.7, for ε > small enough, u ε is a solutionof the initial problem (1). The existence of a sequence ( x ε ) ε> ⊂ Λ such that lim inf ε → u ε ( x ε ) > follows from Lemma 3.2 and the concentration result follows from Proposition 3.7. Finally,Proposition 4.6 yields the decay estimate. (cid:3) ONCENTRATION ON CIRCLES FOR NONLINEAR SCHRÖDINGER-POISSON 19 Remarks and further results Concentration at points. We can also obtain a result about solutions concentrat-ing at points. In this case, the concentration function is given by A ( x ) = [ V ( x )] p +1 p − − [ K ( x )] − p − . Theorem 4. Let < p < , V, K ∈ C ( R \ { } , R + ) , K and ρ ∈ L / loc ( R ) ∩ L ∞ loc (cid:0) R \ { } (cid:1) . Assume that one set ( G i ∞ ) of growth conditions at infinity and one set ( G j ) of growth conditions at the origin hold. Assume also that there exists a smooth openbounded set Λ ⊂ R such that < inf Λ A < inf ∂ Λ A . (41) Then there exists ε > such that for every < ε < ε , problem (1) has at least onepositive solution u ε . Moreover, for every < ε < ε , there exists x ε ∈ Λ such that u ε attains its maximum at x ε , lim inf ε → u ε ( x ε ) > , lim ε → A ( x ε ) = inf Λ A , and there exist C > and λ > such that u ε ( x ) ≤ C exp (cid:18) − λε | x − x ε | | x − x ε | (cid:19) (cid:0) | x − x ε | (cid:1) − , ∀ x ∈ R . The proof of this theorem is similar to the proof of Theorem 2, but simpler. Let us onlysketch the proof. First of all, we impose no symmetry neither on the potentials nor on thesolution. This makes the critical Sobolev exponent to appear, in spite of what happens inthe preceding results. We modify the problem in the same way as before and we searchfor a critical point of the functional J ε in the space X defined by (13). Theorem 3 remainstrue with the same proof.The limiting problem associated to concentration at points is the problem − ∆ u + au = bu p in R . Let x ∈ Λ be a point such that A ( x ) = inf Λ A . We denote by c the least energy criticalvalue of the limiting problem with a = V ( x ) and b = K ( x ) . As in Proposition 3.1, weprove that the critical value c ε defined in (15) satisfies c ε ≤ ε ( c + o (1)) as ε → . Then we prove as before that the L ∞ -norm of u ε does not converge to in Λ and thatthe sequence of rescaled solutions converges in C loc to a solution of the limiting equation.The analogous of Proposition 3.6 is the following one. Proposition 5.1. Let ( ε n ) n ⊂ R + and ( x in ) n ⊂ R be sequences such that ε n → andfor ≤ i ≤ M , x in → ¯ x i ∈ ¯Λ as n → ∞ . If for every ≤ i < j ≤ M , we have lim sup n →∞ | x in − x jn | ε n = ∞ and if for every ≤ i ≤ M , lim inf n →∞ u ε n ( x in ) > , then lim inf n →∞ ε − n c ε n ≥ M X i =1 C (¯ x i ) , where C ( x ) is the least energy critical value of the limiting problem with a = V (¯ x ) and b = K (¯ x ) . The concentration result can be stated as follows. Proposition 5.2. Suppose that Λ satisfies (41) . If ( x ε ) ε> ⊂ Λ is such that lim inf ε → u ε ( x ε ) > , then (i) lim ε → A ( x ε ) = inf Λ A ,(ii) lim inf ε → d ( x ε , ∂ Λ) > ,(iii) for every δ > , there exists ε > and R > such that, for every ε ∈ (0 , ε ) , k u ε k L ∞ (Λ \ B ( x ε ,εR )) ≤ δ. Finally the comparison arguments in order to get back to the original problem are thesame as in section 4.5.2. Concentration on spheres. Using the same method, we can prove the existenceof solutions concentrating on a sphere for the following problem. − ε ∆ u + V ( x ) u + ρ ( x ) φu = K ( x ) u p , x ∈ R , − ∆ φ = ερ ( x ) u . However we are not sure whether this problem has a physical meaning.5.3. Concentration on Keplerian orbits. An interesting question related to [9, 15],concerns the existence of solutions concentrating on Kepler orbits, assuming radial po-tentials. For the reasons described in the Introduction, this might be a typical situationwhere the correspondence principle can be checked using solutions localized on classicalplanar orbits. We wonder if this result could be obtained for the D nonlinear Schrödingerand Schrödinger-Poisson equations with radial potentials.5.4. Concentration driven by ρ . If V ≡ K ≡ , it is natural to ask whether there stillexist solutions with a concentration behaviour. In this case, we expect the location of theconcentration points to be governed by the weight ρ . The asymptotic analysis seems moredelicate since it requires higher order estimates. Acknowledgements The authors would like to thank Professor Antonio Ambrosettifor taking their attention to these questions. C.M. would like to thank the members ofthe department of Mathematics of Université Libre de Bruxelles for the kind hospitalityand friendship. C.M. was partially supported by FIRB Analysis and Beyond and PRIN2008 Variational Methods and Nonlinear Differential Equations. References [1] A. Ambrosetti, On Schrödinger-Poisson systems. Milan J. Math. 76 (2008), 257–274.[2] A. Ambrosetti, V. Felli, and A. Malchiodi, Ground states of nonlinear Schrödinger equations withpotentials vanishing at infinity , J. Eur. Math. Soc. (JEMS) (2005), no. 1, 117–144.[3] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and appli-cations. J. Funct. Anal. 14 (1973), 349–381.[4] V. Benci and T. D’Aprile The semiclassical limit of the nonlinear Schrödinger equation in a radialpotential Journal of Differential equations, 184 (2002) 109-138.[5] D. Bonheure, J. Di Cosmo and J. Van Schaftingen, Nonlinear Schrödinger equation with unboundedor vanishing potentials: solutions concentrating on lower dimensional spheres , preprint.[6] D. Bonheure and C. Mercuri, Embedding theorems and existence for nonlinear Schrödinger-Poissonsystems with unbounded and vanishing potentials , preprint Sissa (2010).[7] D. Bonheure and J. Van Schaftingen, Bound state solutions for a class of nonlinear Schrödingerequations , Rev. Mat. Iberoam. 24 (2008), no. 1, 297–351.[8] T. D’ Aprile, On a class of solutions with non-vanishing angular momentum for nonlinearSchrödinger equations. Differential and integral equations , 3 (2003), 349-384.[9] J.-C. Gay, D. Delande and A. Bommier, Atomic quantum states with maximum localization onclassical elliptical orbits. Phys. Rev. A 39 (1989) 6587–6590.[10] M. del Pino, P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded do-mains. Calc. Var. 4 (1996), 121-137.[11] E.H. Lieb and M. Loss, Analysis . Second edition. Graduate Studies in Mathematics, 14. AmericanMathematical Society, Providence, RI, 2001.[12] C. Mercuri, Positive solutions of nonlinear Schrödinger-Poisson systems with radial potentialsvanishing at infinity , Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.19 (2008), no. 3, 211–227.[13] V. Moroz and J. Van Schaftingen, Bound state stationary solutions for nonlinear Schrödingerequations with fast decaying potentials , Calc. Var. Partial Differential Equations 37 (2010), no. 1,1–27.[14] R. Palais, The principle of symmetric criticality , Comm. Math Phys. 69 (1979), no. 1, 19–30.[15] M. Nauenberg, Quantum wave packet on Kepler elliptic orbits. Phys. Rev. A 40 (1989) 1133–1136.[16] D. Ruiz, On the Schrödinger-Poisson-Slater system: behavior of minimizers, radial and nonradialcases. Preprint.[17] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct.Anal. 237 (2006), no. 2, 655–674. ONCENTRATION ON CIRCLES FOR NONLINEAR SCHRÖDINGER-POISSON 21 [18] M. Willem, Minimax theorems. Progress in Nonlinear Differential Equations and their Applications,24. Birkhäuser Boston, Inc., Boston, MA, 1996. Département de Mathématique, Université libre de Bruxelles, CP 214, Boulevard du Tri-omphe, B-1050 Bruxelles, Belgium E-mail address : [email protected] Département de Mathématique, Université catholique de Louvain, Chemin du Cyclotron 2,1348 Louvain-la-Neuve, BelgiumDépartement de Mathématique, Université libre de Bruxelles, CP 214, Boulevard du Tri-omphe, 1050 Bruxelles, Belgium E-mail address : [email protected] S.I.S.S.A./I.S.A.S., Via Bonomea 265, 34136 Trieste, Italy E-mail address ::