Standing waves for coupled nonlinear Schrodinger equations with decaying potentials

TL;DR

This paper investigates positive solutions to a coupled nonlinear Schrödinger system with decaying potentials, showing solutions concentrate around local minima as the perturbation parameter approaches zero, even when potentials vanish or decay at infinity.

Contribution

It introduces a novel analysis for coupled Schrödinger equations with potentials that may vanish and decay, extending existing results to more general potential behaviors.

Findings

Existence of positive solutions concentrating at potential minima.

Solutions persist even with vanishing and decaying potentials.

Concentration behavior as the perturbation parameter tends to zero.

Abstract

We study the following singularly perturbed problem for a coupled nonlinear Schr\"{o}dinger system: {displaymath} {cases}-\e^2\Delta u +a(x) u = \mu_1 u^3+\beta uv^2, \quad x\in \R^3, -\e^2\Delta v +b(x) v =\mu_2 v^3+\beta vu^2, \quad x\in \R^3, u> 0, v> 0 \,\,\hbox{in $\R^3$}, u(x), v(x)\to 0 \,\,\hbox{as $|x|\to \iy$}.{cases}{displaymath} Here, $a, b$ are nonnegative continuous potentials, and $\mu_1,\mu_2>0$. We consider the case where the coupling constant $\beta>0$ is relatively large. Then for sufficiently small $\e>0$, we obtain positive solutions of this system which concentrate around local minima of the potentials as $\e\to 0$. The novelty is that the potentials $a$ and $b$ may vanish at someplace and decay to 0 at infinity.