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

Abstract

This paper consider that the following semilinear elliptic equation $ \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} $ 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.

Keywords

• semilinear elliptic problem
• variable exponent
• mountain pass lamma
• a priori estimate
• positive radial solution