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

Understand

We prove the existence of positive solutions for the supercritical nonlinear fractional Schr\"odinger equation $(-\Delta)^s u+V(x)u-u^p=0$ in $\mathbb R^n$, with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1), \ n>2s$.

• We show that if $V(x)=o(|x|^{-2s})$ 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.