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

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 $\begin{array}{c}-{\mathrm{\Delta }}_{p}u+V(x)|u{|}^{p-2}u-{\mathrm{\Delta }}_p(|u{|}^{2\alpha })|u{|}^{2\alpha -2}u=\lambda h_1(x)|u{|}^{m-2}u+h_2(x)|u{|}^{q-2}u,\hfill \\ x\in {\mathbb{R}}^N,\hfill \end{array}$ \(\begin{aligned}& -\Delta _{p}u+V(x) \vert u \vert ^{p-2}u-\Delta _{p}\bigl( \vert u \vert ^{2\alpha}\bigr) \vert u \vert ^{2\alpha -2}u= \lambda h_{1}(x) \vert u \vert ^{m-2}u+h_{2}(x) \vert u \vert ^{q-2}u, \\& \quad x\in {\mathbb{R}}^{N}, \end{aligned}\) where ${\mathrm{\Delta }}_{p}u=div({|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u)$ $\Delta _{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ , $1<p<N$ $1< p< N$ , $\lambda \ge 0$ $\lambda \ge 0$ , and $1<m<p<2\alpha p<q<2\alpha p^{\ast }=\frac{2\alpha pN}{N-p}$ $1< m< p<2\alpha p<q<2\alpha p^{*}=\frac{2\alpha pN}{N-p}$ . 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 ${\lambda }_0>0$ $\lambda _{0}>0$ such that Eq. (0.1) admits infinitely many high energy solutions in $W^{1,p}({\mathbb{R}}^N)$ $W^{1,p}({\mathbb{R}}^{N})$ provided that $\lambda \in [0,{\lambda }_0]$ $\lambda \in [0,\lambda _{0}]$ .