Existence of a positive radial solution for semilinear elliptic problem involving variable exponent

This paper consider that the following semilinear elliptic equation

(0.1) $$\{\begin{array}{ll}-\mathrm{\Delta }u=u^{q(x)-1},& \text{in}{B}_1,\\ u>0,& \text{in}{B}_1,\\ u=0,& \text{in}\mathrm{\partial }{B}_1,\end{array}$$

where $B_1$ is the unit ball in ${\mathbb{R}}^N(N\ge 3)$, $q(x)=q(|x|)$ is a continuous radial function satifying $2\le q(x)<2^{\ast }=\frac{2N}{N-2}$ and $q(0)>2$. Using variational methods and a priori estimate, the existence of a positive radial solution for (0.1) is obtained.