Bound state solutions for the supercritical fractional Schrodinger equation

发布时间：2018-10-15 作者：浏览次数：

Speaker:	敖微微副教授	DateTime:	2018年10月16日（星期二）上午10:00-10:50
Brief Introduction to Speaker: 敖微微，武汉大学副教授。		Place:	6号楼2楼报告厅
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.