Existence and nonexistence of least energy positive solutions to critical Schrödinger systems with Hardy potential

We are concerned with the following coupled Schrödinger system with a Hardy potential in the critical case {align*} {cases} -Δ uᵢ-{λᵢ}{|x|^2}uᵢ=|u_i|2^*-2u_i+∑j≠ i³βᵢⱼ|uⱼ|2^*/2|u_i|2^*/2-2u_i, ~x∈ R^N,\\ u_i∈ D1,2(R^N),\,\, N≥ 3,\,\, i=1,2,3, {cases} {align*} where 2^*=2N/N-2 , λᵢ∈ (0,ΛN), ΛN:= (N-2)²/4 , and βᵢⱼ=βⱼᵢ for i ≠ j . By virtue of variational methods, we establish the existence and nonexistence of least energy solutions for the purely cooperative case ( βᵢⱼ > 0 for any i ≠ j ) and the simultaneous cooperation and competition case ( β_i₁j₁ > 0 and β_i₂j₂ < 0 for some (i₁, j₁) and (i₂, j₂) ). Moreover, it is shown that fully nontrivial ground state solutions exist when βᵢⱼ≥0 and N≥5 , but NOT in the weakly pure cooperative case ( βᵢⱼ > 0 and small, i ≠ j ) when $N=3,4$ . We emphasize that this reveals that the existence of ground state solutions differs dramatically between $N=3, 4$ and higher dimensions N≥ 5 . In particular, the cases of N = 3 and N≥ 5 are more complicated than the case of N = 4 and the proofs heavily depend on the dimension. Some novel tricks are introduced for N = 3 and N≥5 .