Nonstandard solutions for a perturbed nonlinear Schrödinger system with small coupling coefficients

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,\\ u_2 V_2(x)u_2 |u_2|^{2p 2}u_2 \beta|u_2|^{p 2}|u_1|^pu_2, \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 $+\infty$ $N 1,2$, $V_1$ $V_2$ belong to $C(\mathbb{R}^N,[0,\infty))$. When $p\ge 2$ $\beta>0$ suitably small, show that problem has family of nonstandard solutions $\{w_{\epsilon} (u^1_{\epsilon},u^2_{\epsilon}):0<\epsilon<\epsilon_{0}\}$ concentrating synchronously at common local minimum $V_2$. All decay rates $V_i(i=1,2)$ are admissible can allow close $0$ in paper. Moreover, location concentration points given by Pohozaev identities. Our proofs based on variational methods penalized technique.