Existence of constrained minimizer for a quadratically coupled Schrödinger systems. Issue 1 (2nd January 2020) - Liang, Zhanping; Liu, Jintao
ABSTRACT: In this paper, we consider the existence of solutions to a quadratically coupled Schrödinger systems − Δ u = μ 1 u + 2 α u   v   i n   R N, − Δ v = μ 2 v + α u 2   i n   R N under the condition ∫ R N u 2 = a, ∫ R N v 2 = b . Here N =1, 2, α > 0 and a, b >0 are fixed. In the systems, μ 1 and μ 2 are unknown. We prove that there exists a solution ( μ 1, μ 2, u ~, v ~ ) to the systems with μ 1 < 0, μ 2 < 0, and u ~, v ~ ∈ C 2 ( R N ) being positive, radially symmetric, and decreasing with r = | x | . Our argument is heavily based on the rearrangement techniques.