Nonlinear Schr\"odinger equations near an infinite well potential

Abstract

The paper deals with standing wave solutions of the dimensionless nonlinear Schr\"odinger equation \label{eq:abs1} i\Phi_t(x,t) = -\Delta_x\Phi +V_\la(x)\Phi + f(x,\Phi), \quad x\in\R^N,\ t\in\R,\tag{$NLS_\la$} where the potential $V_\la:\R^N\to\R$ is close to an infinite well potential $V_\infty:\R^N\to\R$, i. e. $V_\infty=\infty$ on an exterior domain $\R^N\setminus\Om$, $V_\infty|_\Om\in L^\infty(\Om)$, and $V_\la\to V_\infty$ as $\la\to\infty$ in a sense to be made precise. The nonlinearity may be of Gross-Pitaevskii type. A solution of \eqref{eq:abs1} with $\la=\infty$ vanishes on $\R^N\setminus\Om$ and satisfies Dirichlet boundary conditions, hence it solves \label{eq:abs2} i\Phi_t(x,t) &= -\Delta_x\Phi +V_\la(x)\Phi + f(x,\Phi), &&\quad x\in\Om,\ t\in\R \Phi(x,t) &= 0 &&\quad x\in\pa\Om,\ t\in\R. \tag{$NLS_\infty$}. We investigate when a solution $\Phi_\infty$ of the infinite well potential \eqref{eq:abs2} gives rise to nearby solutions $\Phi_\la$ of the finite well potential \eqref{eq:abs1} with $\la\gg1$ large. Considering \eqref{eq:abs2} as a singular limit of \eqref{eq:abs1} we prove a kind of singular continuation type results.