Standing waves for coupled nonlinear Schrodinger equations with decaying potentials

Abstract

We study the following singularly perturbed problem for a coupled nonlinear Schrödinger system: {displaymath} {cases}-\e^2Δu +a(x) u = μ_1 u^3+βuv^2, \quad x\in \R^3, -\e^2Δv +b(x) v =μ_2 v^3+βvu^2, \quad x\in \R^3, u> 0, v> 0 \,\,\hbox{in ${\mathbb{R}}^3$}, 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.

Final version, published in JMP

References in corpus (1)

• Semiclassical stationary states for nonlinear Schroedinger equations with fast decaying potentials