Existence of a Radial Solution for a Kirchhoff Type Problem in an annulus

Abstract

In this paper, by using a minimization principle, we study the existence of a radial solution for the following Kirchhoff type problems: $$\left\{ \begin{array}{c} -(a+b\int_{\Omega}\vert \nabla u\vert^2 dx)\Delta u=\lambda f\left(\vert x\vert,u\right)\ \mbox {in}\ \Omega, \\ u=0\ \mbox{on} \ \partial\Omega, \\   \end{array} \right.$$ where $\lambda >0 $ is a parameter, $\Omega =\lbrace x\in \mathbb{R}^N : \alpha<\vert x\vert <\beta \rbrace$, $a,\ b,\ \alpha,\ \beta >0$, $N\geq 2$, $\Delta$ is the Laplacian operator and $f:\left[ \alpha, \beta\right]\times \mathbb{R}\rightarrow \mathbb{R} $ is a continuous function. We will prove the existence of a radial solution for large values of $\lambda$.