On a class of Kirchhoff type problems involving Hardy type potentials

Abstract

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].