Multiple small solutions for Schrödinger equations involving positive quasilinear term

Abstract

We consider the multiplicity of solutions of a class of quasilinear Schrödinger equations involving the $p$-Laplacian: \begin{equation*} -\Delta_{p} u+V(x)|u|^{p-2}u+\Delta_{p}(u^{2})u=K(x)f(x,u),\qquad x\in \mathbb{R}^{N}, \end{equation*} where $\Delta_{p} u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$, $1<p<N$, $N\geq3$, $V$, $K$ belong to $C(\mathbb{R}^{N})$ and $f$ is an odd continuous function without any growth restrictions at large. Our method is based on a direct modification of the indefinite variational problem to a definite one. Even for the case $p=2$, the approach also yields new multiplicity results.

Keywords

• quasilinear schrödinger equations
• variational methods
• brezis–kato type estimates