A Schr\"odinger singular perturbation problem

TL;DR

This paper investigates the existence and limiting behavior of solutions to a Schrödinger singular perturbation problem in three-dimensional space, focusing on conditions that ensure solutions converge to a solution of a reduced problem as the perturbation parameter approaches zero.

Contribution

The paper establishes conditions on the potential and nonlinearity under which solutions exist and converge to a limiting solution in a Schrödinger singular perturbation setting.

Findings

Existence of solutions under specified conditions.

Convergence of solutions as perturbation parameter tends to zero.

Characterization of the limiting problem.

Abstract

Consider the equation $-\ve^2\Delta u_\ve+q(x)u_\ve=f(u_\ve)$ in $\R^3$, $|u(\infty)|<\infty$, $\ve=const>0$. Under what assumptions on $q(x)$ and $f(u)$ can one prove that the solution $u_\ve$ exists and $\lim_{\ve\to 0} u_\ve=u(x)$, where $u(x)$ solves the limiting problem $q(x)u=f(u)$? These are the questions discussed in the paper.