Infinitely many solutions for $\Delta _\alpha$-Laplaceequations with sign-changing potential

Abstract

In this paper, we use some critical point theorems to discuss the existence of weak solutions for the nonlinear elliptic equations $- \Delta _{\alpha } u + V(x) u = f(x, u)+g(x)|u|^{q-2}u$ and $- \Delta _{\alpha } u + V(x) u = f(x, u) +\lambda u$ in $\Omega$ with $u=0$ on $\partial \Omega$, where $\Omega$ is a bounded domain in $\mathbb R^n$, $\Delta _{\alpha }$ is the strongly degenerate operator, the functions $\alpha =(\alpha _1, \ldots , \alpha _n) : \; \mathbb R^n \rightarrow \mathbb R^n$ satisfies some certain conditions, $\lambda$ is a parameter, $q\in (1,2)$, and $f:\Omega \times \mathbb R\rightarrow \mathbb R$ is a Carathéodory function and g is a function that we will specify later.