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.