Semiclassical states for fractional logarithmic Schrödinger equations

Abstract

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

arXiv comment20 pages