Normalized solutions of two-component nonlinear Schrödinger equations with linear couples

Abstract

ABSTRACT In this paper, we focus on the following nonlinear Schrödinger equations with linear couples { − Δ u + V 1 ( x ) u + λ 1 u = μ 1 ∫ R 3 | u ( y ) | p | x − y | d y | u | p − 2 u + βv in R 3 , − Δv + V 2 ( x ) v + λ 2 v = μ 2 ∫ R 3 | v ( y ) | q | x − y | d y | v | q − 2 v + βu in R 3 , ∫ R 3 | u | 2 d x = a , ∫ R 3 | v | 2 d x = b , where 5 3 < p , q < 7 3 , 0 $ ]]> μ 1 , μ 2 > 0 , a , b ≥ 0 , β ∈ R ∖ { 0 } , λ 1 , λ 2 ∈ R are Lagrange multipliers and V 1 ( x ) , V 2 ( x ) : R 3 → R are trapping potentials. We prove the existence of the solutions with prescribed L 2 ( R ) -norm with trivial trapping potentials and nontrivial trapping potentials by applying the rearrangement inequalities.