Nonstandard solutions for a perturbed nonlinear Schr\"{o}dinger system with small coupling coefficients\protect\thanks{A perturbed nonlinear Schr\"{o}dinger system

TL;DR

This paper constructs nonstandard solutions for a weakly coupled nonlinear Schrödinger system with small coupling, showing they concentrate at local minima of the potentials, using variational and penalization methods.

Contribution

It introduces a new class of solutions concentrating at local minima for small coupling constants in a perturbed Schrödinger system.

Findings

Existence of solutions concentrating at local minima.

Solutions are valid for all decay rates of potentials.

Allows small positive coupling constants close to zero.

Abstract

In this paper, we consider the following weakly coupled nonlinear Schr\"odinger system \begin{equation*} \left\{ \begin{array}{ll} -\epsilon^{2}\Delta u_1 + V_1(x)u_1 = |u_1|^{2p - 2}u_1 + \beta|u_1|^{p - 2}|u_2|^pu_1, & x\in \mathbb{R}^N,\\ -\epsilon^{2}\Delta u_2 + V_2(x)u_2 = |u_2|^{2p - 2}u_2 + \beta|u_2|^{p - 2}|u_1|^pu_2, & x\in \mathbb{R}^N, \end{array} \right. \end{equation*} where $\epsilon>0$, $\beta\in\mathbb{R}$ is a coupling constant, $2p\in (2,2^*)$ with $2^* = \frac{2N}{N - 2}$ if $N\geq 3$ and $+\infty$ if $N = 1,2$, $V_1$ and $V_2$ belong to $C(\mathbb{R}^N,[0,\infty))$. When $p\ge 2$ and $\beta>0$ is suitably small, we show that the problem has a family of nonstandard solutions $\{w_{\epsilon} = (u^1_{\epsilon},u^2_{\epsilon}):0<\epsilon<\epsilon_{0}\}$ concentrating synchronously at the common local minimum of $V_1$ and $V_2$. All decay rates of $V_i(i=1,2)$ are admissible and we can allow that $\beta>0$ is close to $0$ in this paper. Moreover, the location of concentration points is given by local Pohozaev identities. Our proofs are based on variational methods and the penalized technique.