Infinitely many solutions to quasilinear elliptic equation with concave and convex terms

Abstract

In this paper, we are concerned with the following quasilinear elliptic equation with concave and convex terms $$ -\Delta u-{\frac12}u\Delta(|u|^2)=\alpha|u|^{p-2}u+\beta|u|^{q-2}u,\quad x\in \Omega, \leqno(\rom{P}) $$% where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain, $1< p< 2$, $4< q\leq 22^*$. The existence of infinitely many solutions is obtained by the perturbation methods.