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- W2801858322 abstract "We prove the existence of positive solutions for the supercritical nonlinear fractional Schrodinger 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 $sin (0,1), n>2s$. 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. Our main contributions are: the existence of a smooth, radially symmetric, entire solution of $(-Delta)^s w=w^p$ 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." @default.
- W2801858322 created "2018-05-17" @default.
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- W2801858322 date "2018-05-08" @default.
- W2801858322 modified "2023-09-24" @default.
- W2801858322 title "Bound state solutions for the supercritical fractional Schrodinger equation" @default.
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- W2801858322 doi "https://doi.org/10.48550/arxiv.1805.02915" @default.
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