Infinitely many solutions of degenerate quasilinear Schrödinger equation with general potentials

Abstract

In this paper, we study the following quasilinear Schrödinger equation: $-div(a(x,\mathrm{\nabla }u))+V(x)|x{|}^{-\alpha p^{\ast }}|u{|}^{p-2}u=K(x)|x{|}^{-\alpha p^{\ast }}f(x,u)\text{in }{\mathbb{R}}^N,$ where $N\ge 3$, $1<p<N$, $-\mathrm{\infty }<\alpha <\frac{N-p}{p}$, $\alpha \le e\le \alpha +1$, $d=1+\alpha -e$, $p^{\ast }:=p^{\ast }(\alpha ,e)=\frac{Np}{N-dp}$ (critical Hardy–Sobolev exponent), V and K are nonnegative potentials, the function a satisfies suitable assumptions, and f is superlinear, which is weaker than the Ambrosetti–Rabinowitz-type condition. By using variational methods we obtain that the quasilinear Schrödinger equation has infinitely many nontrivial solutions.