Normalized ground states to a cooperative system of Schrödinger equations with generic $L^2$-subcritical or $L^2$-critical nonlinearity

Abstract

We look for ground state solutions to the Schödinger-type system    −∆uj + λjuj = ∂jF (u) ∫ R u2j dx = a 2 j (λj , uj) ∈ R×H1(RN ) j ∈ {1, . . . ,M} with N ≥ 1 and 1 ≤ M < 2 + 4/N , where a = (a1, . . . , aM ) ∈]0,∞[M is prescribed and (λ, u) = (λ1, . . . , λM , u1, . . . uM ) is the unknown. We provide generic assumptions on the nonlinearity F which correspond to the L-subcritical and L-critical cases, i.e., when the energy is bounded from below for all or some values of a. Making use of a recent idea, we minimize the energy over the constraint { |uj |L2 ≤ aj for all j } and then provide further assumptions that ensure |uj |L2 = aj .