Nonlinear Schrödinger equations near an infinite well potential

Abstract

The paper deals with standing wave solutions of the dimensionless nonlinear Schrödinger equation

where the potential $V_\lambda :\mathbb {R}^N\rightarrow \mathbb {R}$ is close to an infinite well potential $V_\infty :\mathbb {R}^N\rightarrow \mathbb {R}$, i. e. $V_\infty =\infty$ on an exterior domain $\mathbb {R}^N\setminus \Omega$, $V_\infty |_\Omega \in L^\infty (\Omega )$, and $V_\lambda \rightarrow V_\infty$ as $\lambda \rightarrow \infty$ in a sense to be made precise. The nonlinearity may be of Gross–Pitaevskii type. A standing wave solution of $(NLS_\lambda )$ with $\lambda =\infty$ vanishes on $\mathbb {R}^N\setminus \Omega$ and satisfies Dirichlet boundary conditions, hence it solves

We investigate when a standing wave solution $\Phi _\infty$ of the infinite well potential $(NLS_\infty )$ gives rise to nearby solutions $\Phi _\lambda$ of the finite well potential $(NLS_\lambda )$ with $\lambda \gg 1$ large. Considering $(NLS_\infty )$ as a singular limit of $(NLS_\lambda )$ we prove a kind of singular continuation type results.