Infinitely many solutions for quasilinear Schrödinger equation with concave-convex nonlinearities

Abstract

In this work, we study the existence of infinitely many solutions to the following quasilinear Schrödinger equations with a parameter α and a concave-convex nonlinearity: 0.1 $$\begin{aligned}& -\Delta _{p}u+V(x) \vert u \vert ^{p-2}u-\Delta _{p}\bigl( \vert u \vert ^{2\alpha}\bigr) \vert u \vert ^{2\alpha -2}u= \lambda h_{1}(x) \vert u \vert ^{m-2}u+h_{2}(x) \vert u \vert ^{q-2}u, \\& \quad x\in {\mathbb{R}}^{N}, \end{aligned}$$ where $\Delta _{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ , $1< p< N$ , $\lambda \ge 0$ , and $1< m< p<2\alpha p0$ such that Eq. (0.1) admits infinitely many high energy solutions in $W^{1,p}({\mathbb{R}}^{N})$ provided that $\lambda \in [0,\lambda _{0}]$ .