Standing waves for coupled nonlinear Schrodinger equations with decaying potentials

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.