Bound state solutions for the supercritical fractional Schrödinger equation

Abstract We prove the existence of positive solutions for the supercritical nonlinear fractional Schrodinger equation ( − Δ ) s u + V ( x ) u − u p = 0 in R n , with u ( x ) → 0 as | x | → + ∞ , where p > n + 2 s n − 2 s for s ∈ ( 0 , 1 ) , n > 2 s . We show that if V ( x ) = o ( | x | − 2 s ) as | x | → + ∞ , then for p > n + 2 s − 1 n − 2 s − 1 , this problem admits a continuum of solutions. More generally, for p > n + 2 s n − 2 s , 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 ) s w = w p in 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, Gonzalez and Wei on the singular fractional Yamabe problem.