Bound state solutions for the supercritical fractional Schrödinger equation

Abstract

We prove the existence of positive solutions for the supercritical nonlinear fractional Schrödinger equation (−Δ)su+V(x)u−up=0inRn, with u(x)→0 as |x|→+∞, where p>n+2sn−2s for s∈(0,1),n>2s. We show that if V(x)=o(|x|−2s) as |x|→+∞, then for p>n+2s−1n−2s−1, this problem admits a continuum of solutions. More generally, for p>n+2sn−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 (−Δ)sw=wpinRn, 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, González and Wei on the singular fractional Yamabe problem.