Bound state solutions for the supercritical fractional Schr\"odinger equation

We prove the existence of positive solutions for the supercritical nonlinear fractional Schr"odinger equation $(-\mathrm{\Delta }{)}^{s}u+V(x)u-u^p=0$ in ${\mathbb{R}}^n$, with $u(x)\to 0$ as $\mathrm{\mid }x\mathrm{\mid }\to +\mathrm{\infty }$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1), \ n>2s$\mathrm{.}Weshowthatif$V(x)=o(|x|^{-2s})$as$|x|\to +\infty$, then for $p>\frac{n+2s-1}{n-2s-1}$, this problem admits a continuum of solutions. More generally, for $p>\frac{n+2s}{n-2s}$, conditions for solvability are also provided. This result is the extension of the work by Davila, Del Pino, Musso and Wei to the fractional case. Our main contributions are: the existence of a smooth, radially symmetric, entire solution of $(-\mathrm{\Delta }{)}^{s}w=w^p$ in $\mathbb R^n$, and the analysis of its properties. The difficulty here is the lack of phase-plane analysis for a nonlocal ODE; instead we use conformal geometry methods together with Schaaf’s argument as in the paper by Ao, Chan, DelaTorre, Fontelos, Gonz'alez and Wei on the singular fractional Yamabe problem.