Number of synchronized solutions for linearly coupled elliptic systems

Abstract In this paper, we consider the following linearly coupled Schrodinger system: ( P e ) − e 2 Δ u + u = u 3 + λ v  in  Ω , − e 2 Δ v + v = v 3 + λ u  in  Ω , u > 0 , v > 0  in  Ω , ∂ u ∂ n = ∂ v ∂ n = 0  on  ∂ Ω , where 0 e 1 is a small parameter, 0 λ 1 is a coupling parameter, Ω is a smooth and bounded domain in R 3 , 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 ( P e ) with O ( e − 3 ) interior spikes for sufficiently small e and some λ near 1. In particular, we also show that the problem ( P e ) has exactly O ( e − 3 ) many positive synchronized solutions.