Uniqueness of positive solutions to some coupled nonlinear schrödinger equations

Abstract

We study the uniqueness of positive solutions of the following coupled nonlinear Schrödinger equations: △u 1 - λ 1u 1 + 1u 13 + βu 1u 22 =0 in ℝ N, △u 2 - λ 2u 2 + 2u 23+βu 21u 2 - 0 in ℝ N, u 1 > 0,u 2 > 0,u 1,u 2ε H 1(ℝ N), where N < 3, λ 1, λ 2, 1, 2 are positive constants and β > 0 is a coupling constant. We prove first the uniqueness of positive solution for sufficiently small β > 0. Secondly, assuming that λ 1 = λ 2, we show that u 1 = u 2√β- 1/ √β - 2when β > max{ 1, 2) and thus obtain the uniqueness of positive solution using the corresponding result of scalar equation. Finally, for N = 1 and λ 1 - λ 2, we prove the uniqueness of positive solution when 0 < β ∉ [min{ 1 , 2), max{ 12} ] and thus give a complete classification of positive solutions.