Nonstandard solutions for a perturbed nonlinear Schrödinger system with small coupling coefficients\protect\thanks{A perturbed nonlinear Schrödinger system

Abstract

In this paper, we consider the following weakly coupled nonlinear Schrödinger system \begin{equation*} \left\{ \begin{array}{ll} -ε^{2}Δu_1 + V_1(x)u_1 = |u_1|^{2p - 2}u_1 + β|u_1|^{p - 2}|u_2|^pu_1, & x\in \mathbb{R}^N,\\ -ε^{2}Δu_2 + V_2(x)u_2 = |u_2|^{2p - 2}u_2 + β|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, $2p\in (2,2^{\ast })$ with $2^{\ast }=\frac{2N}{N-2}$ if $N\ge 3$ 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 $\{w_{\epsilon }=(u_{\epsilon }^1,u_{\epsilon }^2):0<\epsilon <{\epsilon }_0\}$ 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.

34 pages

References in corpus (2)

• Limiting Bourgain-Brezis estimates for systems: theme and variations
• Semi-classical states for the Choquard equation