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

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.