INFINITELY MANY SOLUTIONS FOR A PERTURBED SCHR¨ODINGER EQUATION

. We ﬁnd multiple solutions for a nonlinear perturbed Schr¨odinger equation by means of the so–called Bolle’s method.


1.
Introduction.This note concerns with the elliptic equation where N ≥ 2, V is a potential function on R N , g : R N × R → R is a superlinear, but subcritical, nonlinearity (namely, it satisfies the Ambrosetti-Rabinowitz condition) and f : R N → R is a given function.
When f = 0 the study of equation (1) begins with Rabinowitz's paper [15] and then it has been carried out by several authors (cf.[6] and references therein): even if it has a variational structure, the main problem with classical variational tools is the lack of compactness.Thus, in [15] the existence of a nontrivial solution is shown by using the Mountain Pass Theorem but assuming that V ∈ C 1 (R N , R) is positive and coercive; later on in [6], by means of the Symmetric Mountain Pass Theorem (see [1,Theorem 2.8]), Bartsch and Wang find infinitely many solutions if g is odd in u and V is a positive continuous function such that meas x ∈ R N : V (x) ≤ b < +∞ for all b > 0.
Motivated by the fact that on bounded domains, starting with the pioneer papers [2,3,16,19], it is shown that multiplicity results may persist when the symmetry is destroyed by a perturbation term (see also [10,11,22]), we study (1) for f = 0. Our approach is based on the so-called Bolle's method (cf.[9,10]) and on some ideas in [22].We are only aware of a few previous contributions in this direction: indeed, in [17, Theorem 1.1] (see also [18]) it is proved a multiplicity result for a problem related to ours, provided that the eigenvalues of the involved Schrödinger operator have a suitable growth; on the other hand, in [4] a sharp result is obtained under radial assumptions.
Hereafter, in order to have a variational formulation of the problem and to overcome the lack of compactness, we assume the following conditions: and lim where ) is somehow related to (g 2 ): indeed, by (g 2 ) and direct computations it follows that for any ε > 0 there exists a constant a ε > 0 such that In what follows by a solution we mean a weak solution; classical solutions are found when all the involved functions are smooth enough (e.g, cf.[6]).
Our main result is the following.
Corollary 1.3.Assume that (H 1 ) holds and g(x, u) = |u| p−2 u, with p ∈]2, 2 * [.Then, for all f ∈ L p p−1 (R N ) problem (1) has infinitely many solutions, provided that p ∈]2, p N [, where Condition (H 1 ) on function V is weaker than those used in [6,15], as shown in [17, Proposition 3.1].On the other hand, our set of conditions (H 2 ) is similar to the analogous in [17], even if a comparison between [17, Corollary 1.6] and our Theorem 1.2 can be carried out only when the spectrum of the Schrödinger operator is known (cf.Proposition 3.1).For example, if N = 3 and V (x) = |x| 2 , the corresponding operator in L 2 (R 3 ) admits the sequence of eigenvalues (λ k ) k , with λ k = 2k + 3 (cf.[13, p. 514]).Taking the model nonlinearity, Corollary 1.3 gives infinitely many solutions for (1) if p varies in the range ]2, 5+ [.As usual, for results concerning with problems with broken symmetry, Theorem 1.2 is far from being optimal, since we do not cover the entire subcritical range ]2, 2 * [.In spite of this, when dealing with radial assumptions and N ≥ 3, one finds almost optimal results (cf.[4,5] for unbounded domains and [11,20,21] for bounded ones).
The paper is organized as follows: in Section 2 we recall Bolle's method, then in Section 3 we introduce the variational setting of our problem and prove some technical results; finally, in Section 4 we prove Theorem 1.2.

Notations. Throughout this paper we denote by
• d j , C j positive real numbers, for any j ∈ N.
2. Bolle's perturbation method.In this section we introduce the Bolle's perturbation method firstly stated in [9] but in the version presented in [10] and improved in [12], as the involved functionals are C 1 instead of C 2 .The key point of this approach is dealing with a continuous path of functionals (I θ ) θ∈[0,1] which starts at a symmetric functional I 0 and ends at the "true" non-even functional I 1 associated to the given perturbed problem, so that the critical points of mini-max type of the symmetric map I 0 "shift" into critical points of I 1 .
Throughout this section, let (H, • H ) be a Hilbert space with dual (H , • H ) and we have that (H k ) k is an increasing sequence of finite dimensional subspaces of H. Furthermore, we define and Let us assume that: (A 1 ) I satisfies the following variant of the Palais-Smale condition: converges, up to subsequences; (A 2 ) for all b > 0 there exists , which are Lipschitz continuous with respect to the second variable and such that, if (θ, v) ∈ [0, 1] × H, then (A 4 ) I 0 is even and for each finite dimensional subspace V of H it results with η i as in (A 3 ).
Theorem 2.1.Let I : [0, 1] × H → R be a C 1 path of functionals satisfying assumptions (A 1 ) − (A 4 ).Then, there exists C > 0 such that for all k ∈ N it results: 3. Variational set-up.In this section we present the functional framework of our problem.Firstly, by (2) it makes sense to consider the weighted Sobolev space .
The following proposition (cf. ) is essentially self-adjoint, the spectrum of its self-adjoint extension is an increasing sequence (λ n ) n of eigenvalues of finite multiplicity and where M n denotes the eigenspace corresponding to λ n for every n ∈ N.
Lemma 3.3.Assume that (H 1 ) and (g 1 ) hold.Then, setting Φ : By Lemma 3.3 and standard arguments, the weak solutions of (1) are the critical points of the C 1 functional on In order to apply the Bolle's perturbation method, we define the path of functionals I : [0, 1] × E V → R as follows: Now we verify that, under our main assumptions, the path introduced in (7) satisfies conditions (A 1 ) − (A 4 ) in Section 2. Proposition 3.4.Assume that (H 1 ) − (H 2 ) hold.Then, the family Proof.The proof is organized in four steps.
Step 1.Let ((θ n , u n )) n ⊂ [0, 1] × E V be a sequence such that (5) holds; hence, and where ε n 0 as n → +∞.Therefore, by (g 2 ), Remark 3.2 and the Hölder inequality, it follows that thus the sequence (u n ) n is bounded in E V and (A 1 ) follows by Proposition 3.1 and standard arguments.
by using again the Hölder inequality and Remark 3.2, we get that hence (A 2 ) holds.

direct computations and elementary inequalities give
and inequality (6) holds with therefore (A 3 ) is proved.
Step 4. Finally, let us remark that by (g 4 ) the functional I 0 is even on E V (see (7)); moreover, by (g 3 ) and standard arguments we have that Hence, taking any finite dimensional subspace V of E V , as µ > 2 and all norms are equivalent on V, property (A 4 ) follows.
4. Proof of the main results.Our aim is to apply Theorem 2.1, therefore let us introduce a suitable class of mini-max values for the even functional I 0 .
Denoting by (e k ) k the basis of eigenfunctions in E V found in Proposition 3.1, for any k ≥ 1 let us set and where Γ is as in ( 4) with H = E V .
In order to establish a lower estimate for the sequence (c k ) k , we recall two lemmas, proved in [14, Corollary 2] and [4, Lemma 4.2] respectively.
Taking any W : R N → R, we denote by N − (−∆ + W (x)) the number of the negative eigenvalues of the operator −∆ + W (x) and set W − (x) = min{W (x), 0}.
Proof of Theorem 1.2.By Proposition 3.4, Theorem 2.1 applies, so the proof of our result is complete if we rule out case (b) for k large enough or better, as by ( 8) condition (b) implies with c k as in (10), it is enough to prove that (11) cannot hold for k large enough.
In fact, if we assume that ( 11) holds for all k ≥ k 0 for some k 0 ≥ 1, by [2, Lemma 5.3] it follows that there exist C > 0 and k ∈ N such that On the other hand, by (g 1 ) it follows that

[ 7 ,
Theorems 3.1 and 4.1] and [17, Proposition 3.3]) is crucial to overcome the lack of compactness.Proposition 3.1.Let V : R N → R be such that (H 1 ) holds.Then, for all s