Bound states for logarithmic Schrodinger equations with potentials unbounded below

Abstract

We study the existence and concentration behavior of the bound states for the following logarithmic Schrödinger equation \begin{equation*} \begin{cases} -\varepsilon^2Δv+V(x)v=v\log v^2 \ \ &\text {in}\ \ \mathbb R^N,\\ v(x)\to 0 \ \ &\text {as}\ \ |x|\to\infty, \end{cases} \end{equation*} where $N\ge 1$, $\epsilon >0$ is a small parameter, and $V$ may be unbounded below at infinity with a speed of at most quadratic strength. We show that around various types of local topological critical points of the potential function, positive bound state solutions exist and concentrate as $\epsilon \to 0$.

References in corpus (1)

• Existence of a positive solution for a logarithmic Schrödinger equation with saddle-like potential