Vector Solutions for Linearly Coupled Choquard Type Equations with Lower Critical Exponents

The existence, nonexistence, and multiplicity of vector solutions of the linearly coupled Choquard type equations −Δu+V1xu=Iα∗uN+α/Nuα/N−1u+λv,x∈ℝN,−Δv+V2xv=Iα∗vN+α/Nvα/N−1v+λu,x∈ℝN,u,v∈H1ℝN, are proved, where α∈0,N, N≥3, V1xV2x∈L∞ℝN are positive functions, and Iα denotes the Riesz potential