The existence of ground state solutions for semi-linear degenerate Schrödinger equations with steep potential well

Abstract

In this article, we study the following degenerated Schrödinger equations: \begin{align*} \begin{cases} - \Delta _ { \gamma } u + \lambda V ( x ) u = f(x,u) & \text{in} \ \mathbb{R}^{N} ,\\ { u \in E_{\lambda} } \ , \end{cases} \end{align*} where $\lambda > 0$ is a parameter, $\Delta _ { \gamma }$ is a degenerate elliptic operator, the potential $V(x)$ has a potential well with bottom and the nonlinearity $f(x,u)$ is either super-linear or sub-linear at infinity in $u$. The existence of ground state solution be obtained by using the variational methods.

Keywords

• steep well potential
• mountain pass theorem
• strongly degenerate elliptic operator