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 \begin{equation} -\Delta u-{\frac12}\,u\Delta(|u|^2)=\alpha|u|^{p-2}u+\beta|u|^{q-2}u,\quad x\in \Omega, \tag($\rm P$) \end{equation} 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.