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

Abstract

The existence, nonexistence, and multiplicity of vector solutions of the linearly coupled Choquard type equations $\left\{\begin{array}{c}-\Delta u+V_1\left(x\right)u=\left(I_{\alpha }\ast {\left|u\right|}^{\left(N+\alpha \right)/N}\right){\left|u\right|}^{\alpha /N-1}u+\lambda v,x\in {ℝ}^N,\hfill \\ -\Delta v+V_2\left(x\right)v=\left(I_{\alpha }\ast {\left|v\right|}^{\left(N+\alpha \right)/N}\right){\left|v\right|}^{\alpha /N-1}v+\lambda u,x\in {ℝ}^N,\hfill \\ u,v\in H^1\left({ℝ}^N\right),\hfill \end{array}\right.$ are proved, where $\alpha \in \left(0,N\right)$ , $N\ge 3$ , $V_1\left(x\right)V_2\left(x\right)\in L^{\infty }\left({ℝ}^N\right)$ are positive functions, and $I_{\alpha }$ denotes the Riesz potential.