Existence and concentration of positive solution for a fractional logarithmic Schrödinger equation

Abstract

In this paper, by using the variational methods, we study the existence and concentration of positive solutions for the following fractional logarithmic Schrödinger equation $$ \epsilon^{2s}(-\Delta )^{s} u+V(x)u=u\,\text{log}\,u^{2}, \,\, x\in\mathbb{R}^{N}, $$ where $\epsilon>0$ is a parameter, $N > 2s$, $s\in(0,1)$ and $(-\Delta )^{s}$ is the fractional Laplacian, the potential $V:\mathbb{R}^{N}\rightarrow\mathbb{R}$ is a continuous function satisfying a local assumption. We generalize the result obtained by Alves and Ji [3] for the case $s=1$ to the fractional logarithmic Schrödinger equation.