Vector solutions for linearly coupled Choquard type equations with lower critical exponents (Q2246115)

Summary: The existence, nonexistence, and multiplicity of vector solutions of the linearly coupled Choquard type equations \[ \begin{cases} -\varDelta u + V_1(x) u = (I_\alpha \ast |u|^{(N + \alpha)/N}) |u|^{\alpha/N - 1} u + \lambda v,\;x \in \mathbb{R}^N, \\ -\varDelta v + V_2(x) v = (I_\alpha \ast |v|^{(N + \alpha)/N}) |v|^{\alpha/N - 1} v + \lambda u,\;x \in \mathbb{R}^N, \\ u, v \in H^1 (\mathbb{R}^N), \end{cases} \] are proved, where \(\alpha\in(0, N)\), \(N\geq3\), \(V_1(x) V_2(x)\in L^\infty(\mathbb{R}^N)\) are positive functions, and \(I_\alpha\) denotes the Riesz potential.