Semi-classical standing waves for nonlinear Schrödinger systems

Abstract

For $N \le 3$ and $\beta > 0,$ we consider the following singularly perturbed elliptic system

$$\begin{aligned} \left\{ \begin{array}{rl} \varepsilon ^2\Delta u_1 - W_1(x)u_1 + \mu _1 (u_1)^3 +\beta u_1 (u_2)^2= 0,\ u_1 > 0 &{}\quad \text {in }\mathbf{R}^N,\\ \varepsilon ^2 \Delta u_2 - W_2(x)u_2 +\mu _2 (u_2)^3 +\beta u_2(u_1)^2 = 0,\ u_2 > 0 &{}\quad \text {in }\mathbf{R}^N.\\ \end{array} \right. \end{aligned}$$

There are an enormous number of results for localized solutions of singularly perturbed scalar problems using variational methods or finite dimensional reduction methods. However, there exist no general existence results of localized solutions for elliptic systems. We present some such results here. In the first, by a mini-max characterization for a limiting problem, for small $\varepsilon > 0,$ we show the existence of one bump solutions with a common concentration point of $u_1,u_2$ in a domain O when certain conditions for the limiting problem are satisfied. Typical examples of potentials $W_1,W_2$ satisfying the condition are the following: (1) $W_1,W_2$ have a common non-degenerate critical point in O which share the same stable, unstable directions; (2) for the outnormal n on $\partial O$, $\frac{\partial W_1}{\partial n} > 0, \frac{\partial W_2}{\partial n} > 0$ or $\frac{\partial W_1}{\partial n} < 0, \frac{\partial W_2}{\partial n} < 0$ on $\partial O;$ (3) $\max _{x \in O}W_i(x) >> \max _{x \in \partial O}W_i(x)$ for $i =1,2.$ We also give some nonexistence results for some potentials $W_1,W_2$, not satisfying these conditions, but each $W_1,W_2$ having a structurally stable critical point in O.