Number of synchronized solutions for linearly coupled elliptic systems

In this paper, we consider the following linearly coupled Schrodinger system: {-epsilon(2) Delta u + u = u(3) + lambda v in Omega, -epsilon(2) Delta v + v = v(3) + lambda u in Omega, u > 0, v > 0 in Omega, (P-epsilon) partial derivative u/partial derivative n - partial derivative v/partial derivative n - 0 on partial derivative Omega where 0 < epsilon < 1 is a small parameter, 0 < lambda < 1 is a coupling parameter, Omega is a smooth and bounded domain in R-3, and n is the outer normal vector defined on partial derivative Omega, the boundary of Omega. Motivated by the works of Ao and Wei (2014) and Ao et al. (2013), we use the Lyapunov-Schmidt reduction method to construct a positive synchronized solution of the problem (P-epsilon) with O(epsilon(-3)) interior spikes for sufficiently small epsilon and some lambda near 1. In particular, we also show that the problem (P-epsilon) has exactly O(epsilon(-3)) many positive synchronized solutions. (C) 2021 Elsevier Ltd. All rights reserved.

Automated summary

In this paper, a linearly coupled Schrodinger system is considered, and a positive synchronized solution is constructed using the Lyapunov-Schmidt reduction method for sufficiently small ε and some λ near 1. It is also shown that the problem has exactly O(ε^(-3)) many positive synchronized solutions.