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

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 $ϵ>0$, $\beta \in \mathbb{R}$ is a coupling constant, $2p\in (2,2^{\ast })$ with $2^{\ast }=\frac{2N}{N-2}$ if $N\ge 3$ and $+\mathrm{\infty }$ if $N=1,2$, $V_1$ and $V_2$ belong to $C({\mathbb{R}}^N,[0,\mathrm{\infty }))$. 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.