Multiplicity results for sublinear elliptic equations with sign-changing potential and general nonlinearity

In this paper, we study the following elliptic boundary value problem: {−Δu+V(x)u=f(x,u),x∈Ω,u=0,x∈∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} -\Delta u+V(x)u=f(x, u),\quad x\in \Omega , \\ u=0, \quad x \in \partial \Omega , \end{cases} $$\end{document} where Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega \subset {\mathbb {R}}^{N}$\end{document} is a bounded domain with smooth boundary ∂Ω, and f is allowed to be sign-changing and is of sublinear growth near infinity in u. For both cases that V∈LN/2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V\in L^{N/2}(\Omega )$\end{document} with N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\geq 3$\end{document} and that V∈C(Ω,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V\in C(\Omega , \mathbb {R})$\end{document} with infΩV(x)>−∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\inf_{\Omega }V(x)>-\infty $\end{document}, we establish a sequence of nontrivial solutions converging to zero for above equation via a new critical point theorem.


Introduction
Consider the elliptic boundary value problem where V : → R and f : ×R → R, ⊂ R N is a bounded domain with smooth boundary ∂ .
The semilinear elliptic equation has found a great deal of interest in the past several decades. With the aid of variational methods, existence and multiplicity of nontrivial solutions for problem (1.1) have been extensively studied under various assumptions on the potential V (x) and the nonlinearity f (x, u); see [1, 2, 5, 10, 11, 13, 15-18, 20, 25, 26, 28-32] and the references therein.
It is well known that weak solutions to (1.1) correspond to critical points of the energy functional: (u) = 1 2 |∇u| 2 + V (x)u 2 dx -F(x, u) dx, (1.2) where here and in the sequel F(x, t) := t 0 f (x, s) dx. For the case of inf V > -λ 1 ( ), 0 is a local minimum of , techniques based on the mountain-pass theorem have been well applied; see [1,17,26,30]. When inf V ∈ (-∞, -λ 1 ( )), 0 is a saddle point rather than a local minimum of . Problem (1.1) is indefinite, and the main obstacle is to establish the boundedness of the Palais-Smale sequence for . Under assumptions that V ∈ L N/2 ( ) with N ≥ 3 and f is superlinear near infinity in u, Li and Willem [15] obtained one nontrivial solution via a local linking method; see also Willem [30] via the linking theorem [26]. Later, this result was improved by Jiang and Tang [11] under a weak superquadratic condition introduced by Costa [5,6]; see also [18], where local linking and symmetric mountain-pass theorem were also used. In [31], infinitely many solutions of (1.1) was proved by Zhang and Liu by using the variant fountain theorem established in [33]. With the aid of symmetric mountain-pass lemma, this result was improved and generalized by Qin et al. in [25]; see also [13] for similar results. Recently, under some new superquadratic conditions on f , Tang [29] showed that (1.1) has a ground state solution, as well as infinitely many pairs of solutions provided that V satisfies (V) V ∈ C( , R) and inf V (x) > -∞. The existence of infinitely many nontrivial solutions was obtained by He and Zou [10] for the case that f asymptotically shows linear growth near infinity. For related topics including the case of an unbounded domain, we refer the reader to [3, 4, 6, 7, 9, 12, 14, 16, 19, 21-24, 27, 28] and the references therein.
In [32], Zhang and Tang, via the variant fountain theorem established in [33], first studied the sublinear case provided that V ∈ L N/2 ( ) and f satisfies: , and there exist constants μ ∈ [1, 2) and r 1 > 0 such that |u| 2 = ∞ uniformly for x ∈ , and there exist constants c 1 , r 2 > 0 such that Specifically, the following theorem was established [32]. Inspired by the aforementioned work, in the present paper, we continue to study the sublinear case under the following mild assumptions: (S1) F ∈ C 1 ( × R, R), and there exist constants c > 0 and p ∈ (1, 2) such that Our main result reads as follows. Moreover, the nonlinearity f considered in this paper is allowed to be sign-changing. Hence, Theorem 1.2 extends and complements related results in [16,25,31,32].
Before proceeding to the proof of main result, we give two examples to illustrate our assumptions.

Variational setting and proofs of the main results
As in [29], we introduce the variational framework associated with problem (1.1) under (V) which holds also for the case that V ∈ L N/2 ( ).
Denote by A the self-adjoint extension of the operator - : -∞ ≤ λ ≤ +∞} and |A| be the spectral family and the absolute value of A, respectively, and |A| 1/2 be the square root of |A|. Set U = id -E(0) -E(0-). Then U commutes with A, |A| and |A| 1/2 , and A = U|A| is the polar decomposition of A (see [8,Theorem IV 3.3]). Let E = D(|A| 1/2 ) and For any u ∈ E, it is easy to see that u = u -+ u 0 + u + , where Define an inner product and the corresponding norm where (·, ·) L 2 denotes the inner product of L 2 ( ), · s stands for the usual L s ( ) norm. Lemma 2.1 Let (V) be satisfied. Then, for the inner products (·, ·) and (·, ·) L 2 on E, we have For the case that V ∈ L N/2 ( ), spectrum of A consists of only eigenvalues numbered in -∞ < μ 1 ≤ μ 2 ≤ · · · ≤ μ n ≤ 0 < μ n+1 ≤ · · · → +∞ (counted with multiplicity) with the corresponding system of eigenfunctions {e n } forming an orthogonal basis in L 2 ( ); see [

Lemma 2.3
Suppose that V ∈ L N/2 ( ) or (V) holds. Then E is compactly embedded in L s ( ) for 1 ≤ s < 2 * , and there exists τ s > 0 such that u s ≤ τ s u , ∀u ∈ E. (2.9) By (S1) we have Under (S1) and assumptions of Lemma 2.3, the functional defined by (1.2) is of class C 1 (E, R). Moreover, by virtue of (2.3) and (2.6), one has Before presenting the critical point theorem used in this paper, we give some notions. Let X be a Banach space and I ∈ C 1 (X, R) a functional. A sequence {u n } ⊂ X is called a (PS) sequence (or (PS) c sequence) if The functional I is said to satisfy (PS) condition (or (PS) c condition) if each (PS) sequence (or (PS) c sequence) has a convergent subsequence. A subset A ⊂ X is said to be symmetric if u ∈ A implies that -u ∈ A. For a closed symmetric set A which does not contain the origin, we define a genus γ (A) of A by the smallest integer k such that there exists an odd continuous mapping from A to R k \ {0}. If there does not exist such a k, we define γ (A) = ∞. Moreover, we set γ (∅) = 0. Let k denote the family of closed symmetric subset A of X such that 0 / ∈ A and γ (A) ≥ k. Under V ∈ L N/2 ( ) or (V), it follows from (2.5) or (2.7) that dim(E -⊕ E 0 ) < ∞. We choose an orthonormal basis {ξ j } k 0 j=1 for E -, an orthonormal basis {ξ j } l 0 j=k 0 +1 for E 0 and an orthonormal basis {ξ j } ∞ j=l 0 +1 for E + , where k 0 , l 0 ∈ N and 1 ≤ k 0 < l 0 < ∞. Then {ξ j } ∞ j=1 is an orthonormal basis of E. Define (2.14) Proof of Theorem 1.2 Consider the truncated functional which implies that I is bounded from below and satisfies (PS) c condition with c ≤ 0. Indeed, any sequence {u n } ⊂ E satisfying (2.13) is bounded by (2.16). Passing to a subsequence, we may assume that u n u in E. By Lemma 2.3, u n → u in L s ( ) for s ∈ [1, 2 * ), and u - It follows from (S1) and (2.9) that   Thus I satisfies the (PS) c condition with c ≤ 0. For any fixed l 0 + 1 ≤ k ∈ N, where l 0 = dim(E -⊕ E 0 ) < ∞. By (2.14) and the equivalence of the norms on finite dimensional spaces, there exist constants c k , d k > 0 such that u 2 ≥ c k u , ess sup x∈ u(x) := u ∞ ≤ d k u , ∀u ∈ Y k , (2.21) (S2) implies the existence of a constant r ∈ (0, 1) such that F(x, u) ≥ c -2 k |u| 2 for all |u| ≤ r and a.e. x ∈ . Then, for any u ∈ Y k with u = l k := 2 -1 min{1, rd -1 k }, one has which implies that u ∈ Y k : u = l k ⊂ u ∈ E : I(u) ≤ -1 2 l 2 k . (2.23)