Infinitely many solutions for quasilinear Schrödinger equations under broken symmetry situation

Abstract

In this paper, we study the existence of infinitely many solutions for the quasilinear Schrödinger equations $$ \begin{cases} -\Delta u-\Delta(|u|^{\alpha})|u|^{\alpha-2}u=g(x,u)+h(x,u) &\text{for } x\in \Omega,\\ u=0 &\text{for } x\in \partial\Omega, \end{cases} $$ where $\alpha\geq 2$, $g, h\in C(\Omega\times \mathbb{R}, \mathbb{R})$. When $g$ is of superlinear growth at infinity in $u$ and $h$ is not odd in $u$, the existence of infinitely many solutions is proved in spite of the lack of the symmetry of this problem, by using the dual approach and Rabinowitz perturbation method. Our results generalize some known results and are new even in the symmetric situation.