Bound states for logarithmic Schrodinger equations with potentials unbounded below

TL;DR

This paper investigates the existence and concentration of positive bound states for a logarithmic Schrödinger equation with potentials that may be unbounded below, focusing on solutions that concentrate near critical points as the parameter approaches zero.

Contribution

It establishes the existence and concentration behavior of bound states for the logarithmic Schrödinger equation with unbounded potentials at infinity, expanding understanding of such equations.

Findings

Bound states exist near various local critical points of the potential.

Solutions concentrate as the parameter epsilon approaches zero.

The potential can be unbounded below with at most quadratic growth.

Abstract

We study the existence and concentration behavior of the bound states for the following logarithmic Schr\"odinger equation \begin{equation*} \begin{cases} -\varepsilon^2\Delta v+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 $N\geq 1$, $\varepsilon>0$ 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 $\varepsilon\to0$.