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

Abstract

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 − Δ p u + V ( x ) | u | p − 2 u − Δ p ( | u | 2 α ) | u | 2 α − 2 u = λ h 1 ( x ) | u | m − 2 u + h 2 ( x ) | u | q − 2 u , x ∈ R N , $$\begin{aligned}& -\Delta _{p}u+V(x) \vert u \vert ^{p-2}u-\Delta _{p}\bigl( \vert u \vert ^{2\alpha}\bigr) \vert u \vert ^{2\alpha -2}u= \lambda h_{1}(x) \vert u \vert ^{m-2}u+h_{2}(x) \vert u \vert ^{q-2}u, \\& \quad x\in {\mathbb{R}}^{N}, \end{aligned}$$ where Δ p u = div ( | ∇ u | p − 2 ∇ u ) $\Delta _{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ , 1 0 $\lambda _{0}>0$ such that Eq. (0.1) admits infinitely many high energy solutions in W 1 , p ( R N ) $W^{1,p}({\mathbb{R}}^{N})$ provided that λ ∈ [ 0 , λ 0 ] $\lambda \in [0,\lambda _{0}]$ .

Keywords

• Quasilinear Schrödinger equations
• Dual approach
• High energy solution