Existence and nonexistence of least energy positive solutions to critical Schr\"{o}dinger systems with Hardy potential

We are concerned with the following coupled Schr\"{o}dinger system with Hardy potential in the critical case {equation*} {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} {equation*} where 2^*=2N/N-2, λᵢ∈ (0,ΛN), ΛN:= (N-2)²/4, 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.