Infinitely many solutions to quasilinear Schrödinger equations with critical exponent

Abstract

This paper is concerned with the following quasilinear Schrödinger equations with critical exponent: \begin{equation*}\label{eqS0.1} - \Delta _p u+ V(x)|u|^{p-2}u - \Delta _p(|u|^{2\omega}) |u|^{2\omega-2}u = a k(x)|u|^{q-2}u+b |u|^{2\omega p^{*}-2}u,\qquad x\in\mathbb{R}^N. \end{equation*} Here $\Delta _p u =\mathrm{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator with $1< p < N$, $p^* =\frac{Np}{N-p}$ is the critical Sobolev exponent. $1\le 2\omega<q<2\omega p,$ $a$ and $b$ are suitable positive parameters, $V \in C(\mathbb{R}^N, [0, \infty) ),$ $k\in C(\mathbb{R}^N,\mathbb{R})$. With the help of the concentration-compactness principle and R. Kajikiya's new version of symmetric Mountain Pass Lemma, we obtain infinitely many solutions which tend to zero under mild assumptions on $V$ and $k$