Semi-classical Solutions For Fractional Schrodinger Equations With Potential Vanishing At Infinity

Abstract

We study the following fractional Schrödinger equation \begin{equation}\label{eq0.1} \varepsilon^{2s}(-Δ)^s u + Vu = |u|^{p - 2}u,\ \ x\in\,\,\mathbb{R}^N. \end{equation} We show that if the external potential $V\in C({\mathbb{R}}^N;[0,\infty ))$ has a local minimum and $p\in (2+2s/(N-2s),2_s^{\ast })$, where $2_s^{\ast }=2N/(N-2s),N\ge 2s$, the problem has a family of solutions concentrating at the local minimum of $V$ provided that ${\mathrm{liminf}}_{|x|\to \infty }V(x)|x{|}^{2s}>0$. The proof is based on variational methods and penalized technique. {\textbf {Key words}: } fractional Schrödinger; vanishing potential; penalized technique; variational methods.