Infinitely many solutions for perturbed Kirchhoff type problems

In this paper, we discuss a superlinear Kirchhoff type problem where the non-linearity is not necessarily odd. By using variational and perturbative methods, we prove the existence of infinitely many solutions in the non-symmetric case.

When a = 0, b = 0 in problem (1.1), it reduces to the classic semilinear elliptic problem and the existence of solutions for elliptic equations with zero Dirichlet boundary conditions has been widely studied by variational methods, for example, see [2,16,18] .Further suppose that f ≡ 0, and 1 < p < 2 * − 1, here 2 * = 2N/(N − 2) for N ≥ 3, 2 * = +∞ if N = 1, 2, it is well known that (1.1) has infinitely many distinct solutions {u k } associated with critical values I(u k ) of the functional Email: wwbing2013@126.com W. Wang such that I(u k ) → +∞ as k → ∞.If f ≡ 0 and f (x, u) is not odd in u, symmetry of the functional corresponding the equation is lost and the Symmetric Mountain Pass Theorem cannot be applied.A long standing question is whether the symmetry of the functional is necessary for the existence of infinitely many critical points.Since the 1980s, some mathematicians had been working on this problem for elliptic equations, see Bahri and Berestycki [4],Bahri and Lions [5], Bolle [6], Candela, Salvatore and Squassina [9], Rabinowitz [15], Struwe [17], Tanaka [19] and so on.Researchers gave various conditions guaranteeing the existence infinitely many solution when the symmetry of the problem is broken.Here we list a classical result about the following problem where g ∈ L 2 (Ω).
Theorem 1.1 is a particular case of a more general one due to Bahri and Lions [5].It is not known whether the bound N/(N − 2) is optimal.If Ω = B R is the open ball of radius R > 0 and center 0 in R N (N ≥ 3), and g is a radial function, (1.2) has infinitely many radial solutions for any 1 < p < (N + 2)/(N − 2), see Theorem 1.2 of [8].Whether the conclusion of Theorem 1.1 would still hold for all p up to the Sobolev exponent 2 * − 1 = (N + 2)/(N − 2) for the general function g when N ≥ 3 is a open problem.
When b = 0, (1.1) is called nonlocal because of the presence of the term ( Ω |∇u| 2 dx)∆u, which implies that the equation in (1.1) is no longer a point-wise identity.Kirchhoff type problem received great attention only after Lions [13] proposed an abstract functional analysis framework for the problem, see [1,3,11,14].The nonlocal perturbation causes that the energy functional corresponding the equation has properties different than the case b = 0.There are some works showing that sometime the appearance of the term ( Ω |∇u| 2 dx)∆u is good in some sense, see [20].
The main purpose of the present paper is to show that (1.1) has infinitely many solutions when the exponent p is close to the Sobolev exponent if N = 2, 3. Using Morse indices, we obtain the growth estimate of critical level for the functional without perturbative term.Combining with the Bolle , s Perturbation arguments, we prove the following result.Theorem 1.2.Let (C) hold.Then (1.1) has infinitely many solutions if one of the following conditions is satisfies has infinitely many solutions.
This paper is organized as follows.In Section 2, we present Bolle's Perturbation method which is useful for proving multiplicity results for perturbed problems.In section 3 we apply this result to prove Theorem 1.2.Throughout the paper, the symbols C 1 , C 2 , . . .denote various positive constants whose exact values are not essential to the analysis of the problem.

Bolle's perturbation arguments
In order to apply the method introduced by Bolle [6] for dealing with problems with broken symmetry, we recall the main theorem as stated in [7].The idea is to consider a continuous path of functional starting from a symmetric functional and to prove a preservation result for min-max critical levels in order to obtain critical points for a nonsymmetric functional.
Let H be a Hilbert space equipped with the norm • .Assume that H = H − ⊕ H + , where dim(H − ) < +∞, and let (e k ) k≥1 be an orthonormal base of H + .Consider (H3) There exist two continuous maps (H4) J 0 is even and for each finite dimensional subspace W of H it results

the solution of problem
The following abstract result is due to Bolle, Ghoussoub and Tehrani (for more details, see Theorem 2.2 in [7]).

Proof of main result
Consider the Banach space H = H 1 0 (Ω) with the norm u 2 = Ω |∇u| 2 dx and define the functional where F(x, u) = u 0 f (x, r)dr.It is clear that J 0 is an even functional and the solutions of problem (1.1) are the critical points of J 1 .It is also easily shown that in any finite dimensional subspace of H, Using Young's inequality, we have where ε > 0 and A * is conjugate of A, which follow that for ∀ε > 0, there is C(ε) > 0 such that for all u ∈ H, Lemma 3.1.The functional J θ satisfies PS condition.
Proof.Assume that there exist For sufficiently large n, Using (C) and (3.1), we have for sufficiently large n and some C i > 0, i = 1, 2, 3. Hence, for sufficiently large n where u j := u n j , θ j := θ n j .From (C), we have which follows that u j − u → 0 or u j → 0 ( when a = 0 ).If u j → 0, then u = 0, that is, Using (3.1), we obtain that There exist for some C 3 (d) > 0 and C 4 (d) > 0. And where Proof.Since J θ (u) = 0, we have From (3.1) and (3.2), there exist Hence, where . The proof is completed.
Now let H k be the subspace of H spanned by the first k eigenfunctions of ∆.In order to estimate critical levels c k of J 0 , we need the following classical result.Lemma 3.4 ([12, 19]).Let Ω be a bounded smooth domain in R N (N ≥ 2), let q > 1 with q = N 2 if N ≥ 3, and let V ∈ L q (Ω).Denote by m(V) the number of non-positive eigenvalues of the following eigenvalue problem Then there is a constant C q > 0 such that m(V) ≤ C q Ω |V| q dx.Lemma 3.5.
(1) If N = 3, there exists Ĉ > 0 such that c k ≥ Ĉk Proof.We prove the lemma by using Morse indices.One identifies a cohomotopic family F of dimension k (see Definition 5.1 in [10] ) in such a way that if Noting that where •, • denotes the duality product between H −1 (Ω) and H, one have possesses at least k non-positive eigenvalues.In addition, It is easy to check that (p − 1) < p + 1 < 2 * and V ∈ L (Ω).Applying Lemma 3.4 to V, we have .