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

Title: Bound statesolutions 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 Administrationbuilding,School of Mathematical Sciences, Yuquan Campus

Abstract:

We prove the existenceof positive solutions to the supercritical nonlinear fractional Schrodingerequation $(-/Delta)^s u+V(x)u-u^p=0 /mbox{ in } R^n$, 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 solvabilityare also provided. This result is the extension of (Davila-Del Pino-Musso-WeiJDE 2007) to the fractional case.  Themain contributions for the fractional case are the existence of a smooth,radially symmetric, entire solution of $(-/Delta)^s w=w^p /mbox{ in }R^n$ andthe analysis of its properties. The difficulty here is the lack of phase-planeanalysis for a nonlocal ODE; instead we use conformal geometry methods togetherwith Schaaf's argument.

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