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{document}$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u = u^{q(x)-1}, &\ \ {\mbox{in}}\ \ B_1,\\ u>0, &\ \ {\mbox{in}}\ \ B_1,\\ u = 0, &\ \ {\mbox{in}}\ \ \partial B_1, \end{array} \right. \end{equation} $\end{document}

where $ B_1 $ is the unit ball in $ \mathbb{R}^N(N\geq 3) $, $ q(x) = q(|x|) $ is a continuous radial function satifying $ 2\leq q(x) < 2^* = \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.