Optimal number of solutions for nonlinear coupled Schrodinger systems, part I: Synchronized case

In this paper, we consider the following nonlinear Schrodinger systems {-epsilon(2) Delta u + u = mu(1)u(3) + beta uv(2) in Omega, -epsilon(2) Delta v + v = mu(2)v(3) + beta u(2)v in Omega,(A(epsilon)) u > 0, v > 0 in Omega, partial derivative u/partial derivative nu = partial derivative v/partial derivative nu = 0 on partial derivative Omega, where epsilon > 0, mu(1) > 0, mu(2) > 0, beta is an element of (0, min{mu(1), mu(2)}) boolean OR (max{mu(1), mu(2)}, +infinity), Omega is a bounded domain with smooth boundary in R-3 and nu is the outward unit normal defined on partial derivative Omega, the boundary of Omega. By Lyapunov-Schmidt reduction argument, we prove that there exists epsilon(0) > 0 such that for each 0 < epsilon < epsilon(0) and each integer k satisfying 1 <= k <= delta(Omega)/epsilon(3), (A(epsilon)) has a synchronized solution with k interior spikes, where delta(Omega) is a constant depending only on Omega. Moreover, the upper bound of k is optimal and (A(epsilon)) has exactly O(1/epsilon(3)) many synchronized solutions. (C) 2018 Elsevier Inc. All rights reserved.