Number of synchronized solutions for linearly coupled elliptic systems

Abstract

In this paper, we consider the following linearly coupled Schrödinger system: (Pε)−ε2Δu+u=u3+λv in Ω,−ε2Δv+v=v3+λu in Ω,u>0,v>0 in Ω,∂u∂n=∂v∂n=0 on ∂Ω,where 0<ε<1 is a small parameter, 0<λ<1 is a coupling parameter, Ω is a smooth and bounded domain in R3, and n is the outer normal vector defined on ∂Ω, the boundary of Ω. 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 (Pε) with O(ε−3) interior spikes for sufficiently small ε and some λ near 1. In particular, we also show that the problem (Pε) has exactly O(ε−3) many positive synchronized solutions.