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

We study the following fractional Schr"{o}dinger equation \begin{equation}\label{eq0.1} \varepsilon^{2s}(-\Delta)^s u + Vu = |u|^{p - 2}u,\ \ x\in,,$\mathbb{R}$^N. \end{equation} We show that if the external potential $V\in C({\mathbb{R}}^N;[0,\mathrm{\infty }))$ has a local minimum and $p\in (2 + 2s/(N - 2s), 2^_s)$,where$2^_s=2N/(N-2s),,N\ge 2s$, the problem has a family of solutions concentrating at the local minimum of $V$ provided that ${\mathrm{lim\hspace{0.17em}inf}}_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}V(x)\mathrm{\mid }x{\mathrm{\mid }}^{2s}>0$. The proof is based on variational methods and penalized technique. {\textbf {Key words}: } fractional Schr"{o}dinger; vanishing potential; penalized technique; variational methods.