Existence of normalized solutions for Schrödinger systems with linear and nonlinear couplings

Abstract

Abstract In this paper we study the nonlinear Bose–Einstein condensates Schrödinger system { − Δ u 1 − λ 1 u 1 = μ 1 u 1 3 + β u 1 u 2 2 + κ ( x ) u 2 in R 3 , − Δ u 2 − λ 2 u 2 = μ 2 u 2 3 + β u 1 2 u 2 + κ ( x ) u 1 in R 3 , ∫ R 3 u 1 2 = a 1 2 , ∫ R 3 u 2 2 = a 2 2 , $$ \textstyle\begin{cases} -\Delta u_{1}-\lambda _{1} u_{1}=\mu _{1} u_{1}^{3}+\beta u_{1}u_{2}^{2}+ \kappa (x) u_{2}\quad\text{in }\mathbb{R}^{3}, \\ -\Delta u_{2}-\lambda _{2} u_{2}=\mu _{2} u_{2}^{3}+\beta u_{1}^{2}u_{2}+ \kappa (x) u_{1}\quad\text{in }\mathbb{R}^{3}, \\ \int _{\mathbb{R}^{3}} u_{1}^{2}=a_{1}^{2},\qquad \int _{\mathbb{R}^{3}} u_{2}^{2}=a_{2}^{2}, \end{cases} $$ where a 1 $a_{1}$ , a 2 $a_{2}$ , μ 1 $\mu _{1}$ , μ 2 $\mu _{2}$ , κ = κ ( x ) > 0 $\kappa =\kappa (x)>0$ , β < 0 $\beta <0$ , and λ 1 $\lambda _{1}$ , λ 2 $\lambda _{2}$ are Lagrangian multipliers. We use the Ekeland variational principle and the minimax method on manifold to prove that this system has a solution that is radially symmetric and positive.

Keywords

• Nonlinear Schrödinger systems
• Normalized solutions
• Minimax principle