Existence of normalized solutions for semilinear elliptic systems with potential

Abstract

In this paper, we consider the existence of normalized solutions to the following system: −Δu + V1(x)u + λu = μ1u3 + βv2u and −Δv + V2(x)v + λv = μ2v3 + βu2v in R3, under the mass constraint ∫R3u2+v2=ρ2, where ρ is prescribed, μi, β > 0 (i = 1, 2), and λ∈R appears as a Lagrange multiplier. Then, by a min–max argument, we show the existence of fully nontrivial normalized solutions under various conditions on the potential Vi:R3→R(i=1,2).