Semiclassical solutions for fractional logarithmic Schrödinger equations with potentials unbounded below - An, Xiaoming; Yang, Xian (Topological Methods in Nonlinear Analysis, 2024)
In this paper, we consider the following fractional logarithmic Schrödinger equation\begin{equation*}ε^{2s}(-Δ)^s u + V(x)u=u\log u^2\quad \text{in } \R^N,\end{equation*}where $ε> 0$, $N\ge 1$ and $V(x)\in C\big(\R^N,\R\big)$ is a potential which can be unbounded below at infinity.By considering a new penalization, we show that the problem has a nontrivial solution $u_{ε}$ concentrating at a local minimum of $V$as $ε→ 0$.