Positive solutions for nonlinear Schrödinger equations with deepening potential well

Abstract

Consider the following nonlinear Schrödinger equation:

where $\lambda \ge 0$ is a parameter, $g\in L^{\infty }(R^N)$ vanishes on a bounded domain in $R^N$, and the function $f$ is such that

We are interested in whether problem ($\ast$) has a solution for any given $\alpha ,\lambda \ge 0$. It is shown in [14] and [31] that problem ($\ast$) has solutions for some $\alpha$ and $\lambda$. In this paper, we establish the existence of solution of ($\ast$) for all $\alpha$ and $\lambda$ by using a variant of the Mountain Pass Theorem. Based on these results, we give a diagram in the $(\lambda ,\alpha )$-plane showing how the solvability of problem ($\ast$) depends on the parameters $\alpha$ and $\lambda$.