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

Abstract

AbstractIn 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: $$\begin{aligned}& -\Delta _{p}u+V(x) \vert u \vert ^{p-2}u-\Delta _{p}\bigl( \vert u \vert ^{2\alpha}\bigr) \vert u \vert ^{2\alpha -2}u= \lambda h_{1}(x) \vert u \vert ^{m-2}u+h_{2}(x) \vert u \vert ^{q-2}u, \\& \quad x\in {\mathbb{R}}^{N}, \end{aligned}$$ − 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 , where $\Delta _{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ Δ p u = div ( | ∇ u | p − 2 ∇ u ) , $1< p< N$ 1 < p < N , $\lambda \ge 0$ λ ≥ 0 , and $1< m< p<2\alpha p<q<2\alpha p^{*}=\frac{2\alpha pN}{N-p}$ 1 < m < p < 2 α p < q < 2 α p ∗ = 2 α p N N − p . The functions $V(x)$ V ( x ) , $h_{1}(x)$ h 1 ( x ) , and $h_{2}(x)$ h 2 ( x ) satisfy some suitable conditions. Using variational methods and some special techniques, we prove that there exists $\lambda _{0}>0$ λ 0 > 0 such that Eq. (0.1) admits infinitely many high energy solutions in $W^{1,p}({\mathbb{R}}^{N})$ W 1 , p ( R N ) provided that $\lambda \in [0,\lambda _{0}]$ λ ∈ [ 0 , λ 0 ] .