Nodal solutions for a supercritical semilinear problem with variable exponent

Abstract

In this paper, we are concerned with the following nonlinear supercritical elliptic problem with variable exponent, $$\begin{aligned} {\left\{ \begin{array}{ll} -\,\Delta u=|u|^{2^*+|x|^\alpha -2}u,~&{}\text {in}~ B_1(0),\\ u=0,\quad &{}\text {on} ~\partial B_1(0), \end{array}\right. } \end{aligned}$$ where $$2^*=\frac{2N}{N-2}$$ , $$0<\alpha <\min \{\frac{N}{2},N-2\}$$ , and $$B_1(0)$$ is the unit ball in $$\mathbb {R}^N$$ , $$N\ge 3$$ . For any $$k\in \mathbb {N}$$ , we find, by variational methods, a pair of nodal solutions for this problem, which has exactly k nodal points.