Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum
- Dávila, Juan
- del Pino, Manuel
- Dipierro, Serena
- Valdinoci, Enrico
2010 Mathematics Subject Classification
- 35B44 35J10 35R11
- nonlocal quantum mechanics, Green functions, concentration phenomena
For a smooth, bounded Euclidean domain, we consider a nonlocal Schrödinger equation with zero Dirichlet datum. We construct a family of solutions that concentrate at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case, 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 in the expanding domain.
- Anal. PDE, 8 (2015) pp. 1165--1235.