Multiple solutions to a class of $p$-Laplacian Schr\"{o}dinger equations

Abstract

In this paper, we will prove the existence of infinitely many solutions to the following equation by utilizing the variational perturbation method \begin{equation*} -div(A(x,u)|\nabla u|^{p-2}\nabla u)+\frac{1}{p}A_{t}(x,u)|\nabla u|^{p}+V(x)|u|^{p-2}u=g(x, u), ~~ x\in\mathbb{R}^{N}, \end{equation*} where $2<p<N$, $V(x)$ represents a coercive potential. It should be emphasized that the multiplicity of solutions is unsuitable to be discussed under the framework of $W^{1,p}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$, as the functional corresponding to the above equation does not satisfy the Palais-Smale condition in this space. To this end, we will use the variational perturbation method to construct novel perturbation space and perturbation functional. By relying on the established conclusions regarding the multiplicity of solutions for classical $p$-Laplacian equations, we shall analyze the equation in question. Our result addresses positively the open question from Candela et. al in CVPDE.