Multiple solutions for Kirchhoff equations under the partially sublinear case

In this paper, we prove the infinitely many solutions to a class of sublinear Kirchhoff type equations by using an extension of Clark's Theorem established by Zhaoli Liu and Zhi-Qiang Wang.


Introduction and main results
In this paper we study the existence and multiplicity of solutions for the following Kirchhoff type equations: where a, b are positive constants.
When Ω is a smooth bounded domain in R 3 , the problem has been many papers concerned. Perera and Zhang [1] considered the case where f (x, ·) is asymptotically linear at 0 and asymptotically 4−linear at infinity. They obtained a nontrivial solution of the problems by using the Yang index and critical group. Then, in [1] they considered the cases where f (x, ·) is 4−sublinear, 4−superlinear and asymptotically 4−linear at infinity. By various assumption on f (x, ·) near 0, they obtained multiple and sign changing solutions. Cheng and Wu [3], Ma and Rivera [4] studied the existence of positive solutions of (1.2) and He and Zou [5] obtained the existence of infinitely many positive solutions of (1.2), respectively; Mao and Luan [6] obtained the existence of signed and sign-changing solutions for the problem (1.2) with asymptotically 4−linear bounded nonlinearity via variational methods and invariant sets of descent flow; Sun and Tang [7] studied the existence and multiplicity results of nontrivial solutions for the problem (1.2) with the weaker monotony and 4−superlinear nonlinearity. For (1.2), Sun and Liu [8] considered the cases where the nonlinearity is superlinear near zero but asymptotically 4−linear at infinity, and the nonlinearity is asymptotically linear near zero but 4−superlinear at infinity. By computing the relevant critical groups, they obtained nontrivial solutions via Morse theory. Comparing with (1.1) and (1.2), R 3 in place of the bounded domain Ω ⊂ R 3 . This make that the study of the problem (1.1) is more difficult and interesting. Wu [11] considered a class of Schrödinger Kirchhoff-type problem in R N and a sequence of high energy solutions are obtained by using a symmetric Mountain Pass Theorem. In [12], Alves and Figueiredo study a periodic Kirchhoff equation in R N , they get the nontrivial solution when the nonlinearity is in subcritical case and critical case. Liu and He [13] get multiplicity of high energy solutions for superlinear Kirchhoff equations in R 3 . Li, Li and Shi in [15] proved the existence of a positive solution to a Kirchhoff type problem on R N by using variational methods and cut-off functional technique.
In [9], Jin and Wu in consider the following problem: where constants a > 0, b > 0, N = 2 or 3 and f ∈ C(R N × R, R). By using the Fountain Theorem, they obtained the following theorem.
Theorem A [9] Assume that the following conditions hold: If the following assumptions are satisfied, Recently, the authors obtained an extension of Clark's theorem as follows.
Theorem B [10] Let X be a Banach space, Φ ∈ C 1 (X, R). Assume Φ is even and satisfies the (PS) condition, bounded from below, and Φ(0) = 0. If for any k ∈ N, there exists a k−dimensional subspace X k of X and ρ k > 0 such that sup X k ∩Sρ k Φ < 0, where S ρ = {u ∈ X| u = ρ}, then at least one of the following conclusions holds.
(i) There exists a sequence of critical points {u k } satisfying Φ(u k ) < 0 for all k and u k → 0 as k → ∞.
(ii) There exists r > 0 such that for any 0 < a < r there exists a critical point u such that u = a and Φ(u) = 0.
In this paper, we consider the multiple solutions for Kirchhoff equations under the partially sublinear case by using the Theorem C. Our main result is as follows.
Theorem 1.1 Assume that f satisfies (B 3 ) and the following conditions: Then (1.1) possesses infinitely many solutions {u k } such that u k L ∞ → 0 as k → ∞. Remark 1.1. Throughout the paper we denote by C > 0 various positive constants which may vary from line to line and are not essential to the problem.
The paper is organized as follows: in Section 2, some preliminary results are presented. Section 3 is devoted to the proof of Theorem 1.1.

Preliminary
In this Section, we will give some notations and Lemma that will be used throughout this paper.
Moreover, we denote the completion of C ∞ 0 (R 3 ) with respect to the norm . To avoid lack of compactness, we need consider the set of radial functions as follows: Here we note that the continuous embedding H ֒→ L q (R 3 ) is compact for any q ∈ (2, 6). Define a functional Then we have from (f 1 ) that J 1 is well defined on H and is of C 1 , and It is standard to verify that the weak solutions of (1.1) correspond to the critical points of the functional J 1 .

Proofs of the main result
Proof of Theorem 1.1. Choosef ∈ C(R N × R, R) such thatf is odd in u ∈ R, f (x, u) = f (x, u) for x ∈ R N and |u| < δ/2, andf (x, u) = 0 for x ∈ R N and |u| > δ. In order to obtain solutions of (1.1) we consider Moreover, (3.1) is variational and its solutions are the critical points of the functional defined in H by From (f 1 ), it is easy to check that J is well defined on H and J ∈ C 1 (H 1 (R 3 ), R) (see [? ] for more detail), and Note that J is even, and J(0) = 0. For u ∈ H 1 (R 3 ), Hence, it follows from Lemma 2.1 that We now use the same ideas to prove the (PS) condition. Let {u n } be a sequence in H so that J(u n ) is bounded and J ′ (u n ) → 0. We shall prove that {u n } converges. By (3.2), we claim that {u n } is bounded. Assume without loss of generality that {u n } converges to u weakly in H. Observe that Hence, we have It is clear that I 1 → 0 and I 2 → 0 as n → ∞. In the following, we will estimate I 3 , by using (f 3 ),for any R > 0, Therefore, {u n } converges strongly in H and the (PS) condition holds for J. By (f 2 ) and (f 3 ), for any L > 0, there exists δ = δ(L) > 0 such that if u ∈ C ∞ 0 (B r (x 0 )) and |u| ∞ < δ then K(x)F (x, u(x)) ≥ L|u(x)| 2 , and it follows from Lemma 2.1 that This implies, for any k ∈ N, if X k is a k−dimensional subspace of C ∞ 0 (B r (x 0 )) and ρ k is sufficiently small then sup X k ∩Sρ k J(u) < 0, where S ρ = {u ∈ R 3 | u = ρ}. Now we apply Theorem C to obtain infinitely many solutions {u k } for (3.1) such that Finally we show that u k L ∞ → 0 as k → ∞. Let u be a solution of (3.1) and α > 0. Let M > 0 and set u M (x) = max{−M, min{u(x), M }}. Multiplying both sides of (3.1) with |u M | α u M implies 4a (α + 2) 2 By using the iterating method in [10], we can get the following estimate where ν is a number in (0, 1) and C 1 > 0 is independent of u and α. By (3.3) and Sobolev imbedding Theorem [14], we derive that u k L ∞ (R 3 ) → 0 as k → ∞. Therefore, u k are the solutions of (1.1) as k sufficiently large. The proof is completed.

Acknowledgement
Authors would like to express their sincere gratitude to one anonymous referee for his/her constructive comments for improving the quality of this paper.