EXISTENCE OF SOLUTIONS FOR SINGULARLY PERTURBED SCHR¨ODINGER EQUATIONS WITH NONLOCAL PART

. In the present paper we study the existence of solutions for a nonlocal Schr¨odinger equation where 0 < µ < 3 and 6 − µ 3 < p < 6 − µ . Under suitable assumptions on the potential V ( x ), if the parameter ε is small enough, we prove the existence of solutions by using Mountain-Pass Theorem.

1. Introduction. In this paper we study the existence of semiclassical solutions for the following nonlocal semilinear problem: K(x − y)|u(y)| p dy)|u| p−2 u. (1) Equation (1) is closely related to the following nonlocal Schrödinger equation Here m is the mass of the bosons, is the planck constant, W (x) is the external potential and K(x) is the function which possesses information on the mutual interaction between the bosons. The scattering length α, whose sign determines the type of interactions, negative for repulsive interaction and positive for attractive interaction. It is clear that Ψ(x, t) = u(x)e −iEt solves (2) if and only if u(x) solves equation (1) with V (x) = W (x) − E and ε 2 =

MINBO YANG AND YANHENG DING
where it accounts for the finite-range many-body interactions [6]. Nonlocal nonlinearities have attracted considerable interest as a means of eliminating collapse and stabilizing multidimensional solitary waves.
Here we will consider the attractive case, i.e. the scattering length α = +1. In general, if the response function K(x) is the delta function, then the equation (1) becomes a standard semilinear one The study of existence and concentration of the semiclassical states of Schrödinger equation (3) goes back to the pioneer work [9] by Floer and Weinstein. Assuming that V (x) is a globally bounded potential having a nondegenerate critical point and inf V (x) > 0, Floer and Weinstein [9] considered N = 1, g(u) = u 3 and studied firstly the existence of single and multiple spike solutions based on a Lyapunov-Schmidt reductions. This result was extended to higher dimension and for g(u) = |u| p−2 u by Oh in [16,17]. Since then, equation (3) has attracted the interest of many mathematicians under various assumptions on the potential V (x). In [2] Ambrosetti et al. combined the Lyapuno-Schmit reduction method and variational arguments to study concentration phenomena of the solutions at isolated local minima and maxima of V (x) with polynomial degeneracy. Without assumption of non-degeneracy on critical points of V (x), the existence of (positive) solutions was handled in [20] by Rabinowitz purely via variational methods. In [20], still assuming that inf V (x) > 0, Rabinowitz proved the existence of a positive ground state for any ε > 0 by further assuming that , for all x ∈ R 3 and some a > 0, with strict inequality on a set of positive measure. Using a local variational approach, del Pino and Felmer [18,19] constructed positive solutions concentrating around any topologically nontrivial critical point of the potential V (x). A simple example is that of a local minimum. Assuming for instance Λ ⊂ R 3 is a bounded open set such that Then if inf V (x) > 0 there also exists a positive semiclassical solution. We also refer authors to [5] for the case inf V (x) = 0 and [8] for the case where the potential V (x) was allowed to change sign. If the response function K(x) is a function of Coulomb type, for example 1 |x| , then we arrive at the nonlocal Schrödinger equation, which is also called Choquard-Pekar equation, Many efforts have been made to study the existence of nontrivial solutions for problem (4) with constant ε = 1. In [13], by using critical point theory, P.L. Lions obtained a solution u ∈ H 1 (R 3 ), u ≡ 0. Ackermann [1] proposed an approach to prove the existence of infinitely many geometrically distinct weak solutions to the problem with potential V (x) periodic in x i and 0 is not in the spectrum of −∆ + V (x). In [15] Menzala also applied variational method to show the existence of a nontrivial weak solution of the nonlinear Schrödinger equation. For other nonlocal problems, we also want to mention the Schrödinger-Maxwell system which has also been widely considered. For example, Ruiz [21] considered By reducing the system into a single nonlocal Schrödinger equation and work in the radial functions subspace of H 1 (R 3 ), the author is able to obtain the existence results of (5) depending on the parameter p and µ > 0. When ε → 0 i.e. the semiclassical case is involved, the existence and concentration phenomena of solutions for equation (5) have also been deeply studied, see [10,11] and the references therein. The arguments there depend greatly on the Lyapuno-Schmit reduction method and the non-degeneracy property of the ground state solutions of existence and concentration phenomena of solutions for equation (5) have been deeply studied. However there is few works about the existence of semiclassical solutions for equation (4), since little is known about the ground states of the corresponding autonomous limit problem Recently Wei and Winter [23] proved the non-degeneracy property of the ground state solution of (7), and then they studied the existence of multi-bump solutions for equation (4) under the assumptions that inf The aim of the present paper is to continue to study the existence of solutions for the nonlocal Schrödinger equation with small parameter ε where 0 < µ < 3 and 6−µ 3 < p < 6 − µ. In order to obtain the existence of solutions, we will apply variational methods. Under suitable assumptions on the potentials we prove that for small ε, there is at least one ground state solution u ε for (8). To establish the existence results, we assume that the potential V (x) satisfy The main results of this paper are In general, the solutions of (1) might change sign. The following result shows that, under further restrictions, the problem possesses solutions which change sign. Recall that a map θ : R 3 → R 3 is called an orthogonal involution if θ = I and θ 2 = I where I denotes the identity map in R 3 . (8) has at least one solution u ε changing sign and satisfying Let E be a real Banach space and I : E → R a functional of class C 1 . We say that (u n ) ⊂ E is a Palais-Smale ((P.S) for short) sequence at c for I if (u n ) satisfies (i) I(u n ) → c and (ii) I (u n ) → 0, as n → ∞. I satisfies the (P.S) condition at c, if any (P.S) sequence at c possesses a convergent subsequence.
This paper is organized as follows. In Sect.2, we introduce the variational framework and restate the problems in equivalent forms. In Sect.3, we analysis the behaviors of the bounded (P.S) sequences. In Sect.4, we prove the existence of semiclassical solutions for the nonlocal Schrödinger equation (8) by using Mountain-Pass Theorem.
2. Notations and variational framework. In this paper we use C, C i to denote positive constants and B R the open ball centered at the origin with radius R > 0.
and L s (R 3 ), 1 ≤ s ≤ ∞, denotes the Lebesgue space with the norms The best Sobolev constant S is defined by: To prove the existence of semiclassical solutions of (8) for small ε, we may rewrite (8) in an equivalent form, let λ = ε −2 , (8) reads then as for λ → ∞. To find solutions of problem (9) we will apply variational methods. To this end, we introduce the Hilbert spaces and the associated norms u 2 = (u, u). Obviously, it follows from (P 1 ) that E embeds continuously in H 1 (R 3 ) (see [7,22]). Note that the norm · is equivalent to · λ deduced by the inner product In order to investigate the problems in suitable variational framework, we use A λ := −∆+λV in L 2 (R 3 ) to denote the selfadjoint operator related to the Schrödinger operator. By σ(A λ ), σ e (A λ ) and σ d (A λ ) we denote the spectrum, the essential spectrum and the eigenvalues of A λ below λ e := inf σ e (A λ ), respectively. Note that each µ ∈ σ d (A λ ) is of finite multiplicity. The following two Lemmas are proved in [8], we sketch the proofs here for the completeness of the paper.
. Suppose that the assumption (P 1 ) is satisfied, then there holds λ e ≥ λb.
Fix in the following a number b close to b with and k λ be the number of the eigenvalues of A λ which is smaller than λb . We write η λ,j and h λ,j (1 ≤ j ≤ k λ ), for the eigenvalues and eigenfunctions and set L d λ = Span{h λ,1 , · · · , h λ,k λ }. We will also use the following orthogonal decomposition orthogonal with respect to (·, ·) L 2 and (·, ·) λ . From Lemma 2.1, we have λb |u| 2 2 ≤ u 2 λ for all u ∈ E e λ .  (12), for all u ∈ E e λ , the conclusion then follows. The following inequality will be frequently used to study the nonlocal problems.

Let us set
from 0 < µ < 3, 6−µ 3 < p < 6 − µ and Proposition 2.3, we know the energy functional I λ (u) is well defined and belongs to C 1 (E, R). Consequently, in order to obtain solutions of (9), we only need to look for critical points of the energy functional I λ (u). Now the existence results can be restated as (9) has at least one ground state solution u λ satisfying Then if 4 < p + µ < 6 we have u λ → 0 as λ → ∞.
We will use the following standard Mountain Pass Theorem.

Theorem 2.5 ([3])
. Let E be a real Banach space and I : E → R a functional of class C 1 . Suppose that I(0) = 0 and: . there is e with e > r such that I(e) ≤ 0.
Then I possesses a (P.S) c sequence with c ≥ ρ > 0 given by 3. Behavior of the (P.S) sequences. In this section we will analysis the behaviors of the (P.S) sequences of the functional I λ .
Lemma 3.1. Suppose that the assumption (P 1 ) holds. For fixed λ ≥ 1, let (u n ) be a (P.S) c -sequence for I λ . Then c ≥ 0 and (u n ) is bounded in E.
Proof. The conclusion simply follows from the fact that Hence, without loss of generality, we may assume u n u in E and L 2 (R 3 ), u n → u in L s loc (R 3 ) for 1 ≤ s < 2 * , and u n (x) → u(x) a.e. for x ∈ R 3 . Clearly u is a critical point of I λ .
In the following we will utilize the decomposition (11): Write w n := u n − u and decompose w n by w n = w d n + w e n , with w d n ∈ E d λ and w e n ∈ E e λ . From w n 0 it is easy to see w d n → 0 since dim(E d λ ) < ∞. And also u n → u if and only if w n → 0. Lemma 3.3. Suppose that the assumption (P 1 ) holds. There is a constant α 0 > 0 independent of λ such that, for any (P.S) c sequence (u n ) for I λ with u n u, either u n → u along a subsequence or Proof. Assume (u n ) has no convergent subsequence, then lim inf n→∞ w n λ > 0. Lemma 3.2 implies that along a subsequence, one has It follows that i.e.
Since I λ (w n ) → 0, using the the Hardy-Littlewood-Sobolev inequality again, we know with α 0 > 0 independent of λ, proving the Lemma.
From Lemma 3.3, we have the following convergence criterion for the (P.S) sequences.

4.
Proof of the main results. In the following we will prove that I λ satisfies the geometry conditions of the classical Mountain Pass Theorem. And then we can construct small minimax values for I λ at levels where the (P.S) condition holds if the parameter λ is large enough.
Proof. In fact, for all fixed ϕ satisfying Let us define ϕ t = t 6−µ 2p ϕ(tx), t > 0, then we have Since 6 − µ > p, we know as t → 0. Thus the proposition is proved.
Proof. (1). First note that, for each fixed λ, I λ (0) = 0. By the the Hardy-Littlewood-Sobolev inequality, we know Since p > 1, the conclusion follows if u λ is small enough.
Remark 1. In general, since p > 1, to prove (2) of lemma 4.1, we only need to notice that, for any u ∈ E, I λ (tu) → −∞ as t → ∞. Here the purpose of choosing e λ is to construct small Minimax values below the threshold where (P.S) condition holds.