Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum

By J. D'avila, Manuel del Pino, Serena Dipierro, Enrico Valdinoci. 2014

143 citations recorded by the source corpus.

For a smooth, bounded domain $\Omega$, $s\in(0,1)$, $p\in \left(1,\frac{n+2s}{n-2s}\right)$ we consider the nonlocal equation $$ \epsilon^{2s} (-\Delta)^s u+u=u^p \quad {\mbox{in}}\Omega $$ with zero Dirichlet datum and a small parameter $\epsilon>0$. We construct a family of solutions that concentrate as $\epsilon \to 0$ at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case $s=1$, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due to the nonlocal effect. Delicate analysis is needed to overcome the lack of localization, in particular establishing the rather unexpected asymptotics for the Green function of $ \epsilon^{2s} (-\Delta)^s +1$ in the expanding domain $\epsilon^{-1}\Omega$ with zero exterior datum.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ad7b902356e9d58b027108015247bc93f0a28b6c

Citations in the corpus

• Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth (2017, unknown): “Notice that the arguments of [13,15,16,21] heavily rely on the uniqueness and non-degeneracy of ground state solutions to the corresponding autonomous problems which have been proved in [22,23].”