Semiclassical states for fractional logarithmic Schr\"{o}dinger equations

TL;DR

This paper studies positive solutions to a fractional logarithmic Schr"odinger equation, showing solutions concentrate at local minima of the potential without decay restrictions, using a penalization method.

Contribution

Introduces a penalized function approach to prove existence and concentration of solutions for fractional logarithmic Schr"odinger equations with minimal assumptions on potential decay.

Findings

Solutions concentrate at local minima of V as epsilon approaches zero

No decay restrictions on the potential V, including compact support

Established existence of positive solutions using penalization techniques

Abstract

In this paper, we consider the following fractional logarithmic Schr\"odinger equation \begin{equation*} \varepsilon^{2s}(-\Delta)^s u + V(x)u=u\log |u|^2\ \ \text{in}\ \R^N, \end{equation*} where $\varepsilon>0$, $N\ge 1$, $V(x)\in C(\R^N,[-1,+\infty))$. By introducing an interesting penalized function, we show that the problem has a positive solution $u_{\varepsilon}$ concentrating at a local minimum of $V$ as $\varepsilon\to 0$. There is no restriction on decay rates of $V$, especially it can be compactly supported.