NON-DEGENERACY OF SYNCHRONIZED VECTOR SOLUTIONS FOR WEAKLY COUPLED NONLINEAR SCHRODINER SYSTEMS

Y In this paper, we consider the following two-component Schrodiner system {-Delta u(1) + V-1(x)u(1) = mu(1)u(1)(3) + beta u(1)u(2)(2) in R-3, -Delta u(2) + V-2(x)u(2) = mu(2)u(2)(3) + beta u(1)(2)u(2) in R-3, First, we revisit the proof of the existence of an unbounded sequence of non-radial positive vector solutions of synchronized type obtained in S. Peng and Z. Wang [Segregated and synchronized vector solutions for nonlinear Schrodinger systems, Arch. Rational Mech. Anal. 208 (2013), 305-339] to give a point-wise estimate of the solutions. Taking advantage of these estimates, we then show a non-degeneracy result of the synchronized solutions in some suitable symmetric space by use of the locally Pohozaev identities. The main difficulties of BEC systems come from the interspecies interaction between the components, which never appear in the study of single equations. The idea used to estimate the coupling terms is inspired by the characterization of the Fermat points in the famous Fermat problem, which is the main novelty of this paper.