Multiple solutions for asymptotically linear Schro¨dinger equation with sign-changing potentials

In this paper,we study the multiple solutions for a class of asymptotically linear Schr¨odinger equations (−∆u + V (x)u = f(x,u), x ∈ RN, u(x) → 0,as |x| → ∞, where the potential V (x) is allowed to be sing-changing, and f(x,u) satisfies the asymptotically linear condition.

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Introduction

Consider the following asymptotically linear Schro¨dinger equation (-∆u + V (x)u = f(x,u) x ∈ RN, (1.1) u(x) → 0,as |x| → ∞, where V : R N → R, and f : R N × R → R. The multiplicity solutions for (1.1) have been extensively investigated in many papers with the aid of variational methods. For instance, see [], [], [], and the references therein. In the present paper, we will further study the multiplicity solutions for (1.1) where V (x) is allowed to be sign-changing, and f(x,u) satisfies asymptotically linear condition. Our precise assumptions on V and f are the following:

) and infRN V > -∞; (V2) For any h > 0, there exists a contant d0 such that , where meas(•) denotes the Lebesgue measure in R N ; (f1) F(x,u) ≥ 0, F(x,0) = 0 and f(x,u) = o(|u|) , u → 0 uniformly on R N . |f(x,u)| ≤ Cf(|u|) for some Cf > 0, where It is well known that the Schr¨odinger operator -∆+V is selfadjoint and semi bounded in L 2 (R N ). We denote by T the selfadjoint extension of the operator -∆+V with domain D(T) ⊂ L 2 (R). Let {S(λ) : -∞ < λ < +∞} and |T| be the spectral resolution and absolute value of T respectively, and be the square root of |T| with domain Set U = I -S(0) -S(-0), where I is the identity map on L 2 . Then U commutes with T, |T| and , and T = U|T| is the polar decomposition of T.

Let

) and define on S the inner product and norm where (•,•)2 denotes the inner product in L 2 , then S is a Hilbert space.

In the following, let σ(T), σd(T), σe(T)be the spectrum of T, the discrete spectrum of T and the essential spectrum of T respectively. In order to learnt about the spectrum fo T, we first need the following lemma. We will introduce on S the following inner product and norm:

Remark 1.3. The norms kuk and kuk0 on S are equivalent, and for any 2 ≤ p ≤ 2 * , there exists βp > 0 such that kukp ≤ βpkuk,∀u ∈ S .

(1.2) where | • |p is the norm on L p . Let ( 1 . 3 ) be the quadratic form associated with T. Moreover, using the orthogonal projections with respect to the inner product (•,•), we can define a(u,u) = ((p + -p -)u,u) = ku + k 2 -ku -k 2 .

(1.4) for all u = u -+ u 0 + u + ∈ S, where p ± : S → S ± are orthogonal projection operator. We further assume the following hypotheses for f(x,u):

From the above spectral result of the operator T, the set σ(T)∩(0,a0) consistes of only eigenvalues of finite mulitiplicity, where a0 is defined in f3. Let l denote the number of eigenvalues (counted with multiplicity) lying in (0,a0). ). From this we deduce, using (2.4), that

we claim that v 6= 0. Arguing by contradiction we assume that v = 0, choose a > 0 in lemma 2.1, such that 1, where CF is the constant in (f1). Since S a-⊂ S in lemma 2.1 of finite-dimension, then the compactness of the orthogonal projection P a-:→ S a-⊂ S implies . It follows from ( 2 We claim that v + 6= 0. Since F(x,z) ≥ 0, if v + = 0, it follows from (2.8) and (f1) that kvn -k → 0 and thus vn → v = v 0 . Also 0, and vn → v in L 2 .

Note that by (f1) and ( ) and kf(x,u)k ≤ Cfkuk 2 for some Cf > 0, f(x,u)/kuk 2 → 0 as kuk → ∞, ∀x ∈ R N . Since ku(x)k → ∞ if v(x) 6= 0, we obtain (2.9) The last equality holds by vn → v in L 2 and Lebesgue's Dominated Convergence Theorem. Also, by (f3), .

(2.10) From 2.9, 2.10 and since 0 it follows that |vn|2 → 0. Due to dim(S ˜) < ∞, 1 = kvnk → 0 and this contradiction implies that v + 6= 0. Note that (f3) and (??) implies that Z kv + k 2 -kv -k 2 -(a(x)v,v)dx ≤ kv + k 2 -kv -k 2 -a0|v|

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