The radial solutions for a class of Kirchhoff type equation with variable exponent

This paper is concerned with the following Kirchhoff type equation\begin{equation*}\label{eq0.1}\left\{\begin{array}{ll}-(a+b\int_{B}|\nabla u|^{2}dx)\Delta u=u^{q(x)-1}, &\ \ \mbox{in}\ \ B,\\ u\geq0, &\ \ \mbox{in}\ \ B,\\ u=0, &\ \\mbox{in}\ \ \partial B, \end{array} \right.\end{equation*}where $a\geq0$, $b>0$ are real numbers and $B$ is the unit ball in $\R^3$, $q(x)=q(|x|)$ is a continuous radial function satifying $2<\mathop {\min }\limits_{x \in \overline{B}}q(x)=q_-<4 4$. By means of variational methods and a priori estimate, we prove the existence and multiplicity of radial solutions to this problem. Mathematics Subject Classification 2010: 35J20, 35J62, 35Q55.