On a class of Kirchhoff type problems involving Hardy type potentials

This article deals with the multiplicity of solutions for the following Kirchhoff type problem $$ \begin{cases} \begin{array}{rlll} -M\left(\int_\Omega |\nabla u|^2\,dx\right)\Delta u &= &  \frac{\mu}{|x|^2}a(x)u + \lambda f(u) & \text{ in } \Omega,\\ u & = & 0 & \text{ on } \partial\Omega, \end{array} \end{cases} $$ where $\Omega\subset\R^N$ $(N\geq 3)$ is a bounded domain with smooth boundary $\partial\Omega$, $0\in\Omega$, $M:\R^+_0 \to \R$ is a continuous and increasing function, $a:\Omega \to \R$ may change sign, $f:\R\to\R$ is continuous and sublinear at infinity, $\lambda,\mu$ are two parameters. Our proof is based on the three critical points theorem in [3].


Contents 1 Introduction and Preliminaries 289
2 Multiple solutions 291

Introduction and Preliminaries
In this article, we are concerned with a class of Kirchhoff type problems with Hardy type potential where Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary ∂Ω, 0 ∈ Ω, M : R + 0 → R is a continuous and increasing function with R + 0 := [0, +∞), the function a : Ω → R may change sign, λ is a positive parameter, 0 ≤ µ < µ ⋆ , where is the best constant in the Hardy inequality, i.e., for all ϕ ∈ C ∞ 0 (Ω), see [8].Since the first equation in (1.1) contains an integral over Ω, it is no longer a pointwise identity; therefore it is often called nonlocal problem.This problem models several physical and biological systems, where u describes a process which 2000 Mathematics Subject Classification: 34B27, 35J60, 35B05.

N.T. Chung
depends on the average of itself, such as the population density, see [4].Moreover, problem (1.1) is related to the stationary version of the Kirchhoff equation presented by Kirchhoff in 1883, see [7].This equation is an extension of the classical d'Alembert's wave equation by considering the effects of the changes in the length of the string during the vibrations.The parameters in (1.3) have the following meanings: L is the length of the string, h is the area of the cross-section, E is the Young modulus of thematerial, ρ is themass density, and P 0 is the initial tension.
In recent years, problems involving Kirchhoff type operators have been studied in many papers, we refer to [1,2,6,9,12,14,16], in which the authors have used different methods to get the existence of solutions for (1.1) in the case µ = 0.In [11,17], Z. Zhang et al. studied the existence of nontrivial solutions and signchanging solutions.In [5,8,10,13,15] the authors studied the existence of solutions for (1.1) in the case M (t) ≡ 1 and a(x) ≡ 1. Motivated by the papers mentioned above, in this work, we study the existence of solutions for Kirchhoff type problem (1.1) in which the function f is assumed to be sublinear at infinity.Our situation here is different from [1,2,16] in which the authors considered problem (1.1) in the case µ = 0 and f is superlinear or assymptotically linear at infinity.
It should be noticed that by the presence of the singular potential we cannot obtain the similar result for the p-Laplacian −∆ p u.This comes from the fact that the energy functional does not satisfy the Palais-Smale condition.Theorem 1.2 will be proved by using a recent result on the existence of at least three critical points by G. Bonanno [3].For the reader's convenience, we describe it as follows.
) be a separable and reflexive real Banach space, A, F : X → R be two continuously Gâteaux differentiable functionals.Assume that there exists x 0 ∈ X such that A(x 0 ) = F(x 0 ) = 0, A(x) ≥ 0 for all x ∈ X and there exist x 1 ∈ X, ρ > 0 such that and assume that the functional A− λF is sequentially weakly lower semicontinuous, satisfies the Palais-Smale condition and Then, there exist an open interval Λ ⊂ [0, a] and a positive real number δ such that each λ ∈ Λ, the equation DA(u) − λDF(u) = 0 has at least three solutions in X whose .-norms are less than δ.

Multiple solutions
Let us define the functional J µ,λ : H 1 0 (Ω) → R by the following formula

N.T. Chung
where (2.2) In the rest of this paper, we denote by S q the best constant of the embedding Lemma 2.1.There exists µ > 0 such that for any µ ∈ [0, µ), the functional J µ,λ is sequentially weakly lower semicontinuous on H 1 0 (Ω).
Proof: Let {u m } be a sequence that converges weakly to u in H 1 0 (Ω).By the conditions (A) and (M 0 ), taking µ = µ ⋆ m0 A0 , then for each 0 ≤ µ < µ, using the same arguments as in the proof of [10, Theorem 3.2], we can obtain On the other hand, by (F 1 ), there exists a constant C 1 > 0, such that From (2.4) and the Hölder inequality, we get which shows that lim From relations (2.3) and (2.6), we conclude that and thus, the functional J µ,λ is sequentially weakly lower semi-continuous in H 1 0 (Ω).✷ Lemma 2.2.For every µ ∈ [0, µ) and λ ∈ R, the functional J µ,λ is coercive and satisfies the Palais-Smale condition.
Next, let {u m } be a sequence in H 1 0 (Ω), such that where H −1 (Ω) is the dual space of H 1 0 (Ω).Since J µ,λ is coercive, the sequence {u m } is bounded in H 1 0 (Ω).Then, there exists a subsequence of {u m }, still denoted by {u m }, that converges weakly to some u ∈ H 1 0 (Ω) and {u m } converges strongly to u in L 2 (Ω).We will prove that for any u, v ∈ H 1 0 (Ω), where m 0 is given by (M 0 ) and Indeed, using the Cauchy inequality we have Hence, because M (t) is increasing, it implies that (2.12) Now, from (2.9), (2.10) and the Hardy inequality, we find that (2.13) Kirchhoff type problem

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On the other hand, by the Hölder inequality, which approaches 0 as m → ∞.From (2.13), (2.14) and the fact that 0 ≤ µ < µ = µ * m0 A0 , we deduce that u m converges strongly to u in where the functionals A and F are given by (2.2).