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

编辑:wfy 时间:2018年11月06日 访问次数:

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

Speaker: Professor . Ao Weiwei  (Wuhan University)

Time: 2018-11-12  15:45-16:45

Location: 200-9,  Sir  Run Run Shaw Business Administration building,School of Mathematical Sciences, Yuquan Campus


Abstract:

We prove the existence of positive solutions to the supercritical nonlinear fractional Schrodinger equation $(-\Delta)^s u+V(x)u-u^p=0 \mbox{ in } R^n$, with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1), \ 2s\frac{n+2s-1}{n-2s-1}$, 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 (Davila-Del Pino-Musso-Wei JDE 2007) to the fractional case.  The main contributions for the fractional case are the existence of a smooth, radially symmetric, entire solution of $(-\Delta)^s w=w^p \mbox{ in }R^n$ 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.


Contact Person: WANG Meng, (mathdreamcn@zju.edu.cn)