A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system

We study the set of solutions of the nonlinear elliptic system {P} {cases} −Δ u + λ ₁u = μ ₁u³ + β v²u&\:in Ω, \\ −Δ v + λ ₂v = μ ₂v³ + β u²v&\:in Ω , \\ u,v > 0\:in Ω , \:u = v = 0&\:on ∂ Ω , {cases} in a smooth bounded domain Ω ⊂ RN , N⩽3 , with coupling parameter β ∈ R . This system arises in the study of Bose–Einstein double condensates. We show that the value β = −√{μ ₁μ ₂} is critical for the existence of a priori bounds for solutions of (P). More precisely, we show that for β > −√{μ ₁μ ₂} , solutions of (P) are a priori bounded. In contrast, when λ ₁ = λ ₂ , μ ₁ = μ ₂ , (P) admits an unbounded sequence of solutions if β ⩽−√{μ ₁μ ₂} .