Positive ground states for nonlinearly coupled Choquard type equations with lower critical exponents

Abstract

We study the coupled Choquard type system with lower critical exponents $$ \textstyle\begin{cases} -\Delta u+\lambda _{1}(x)u=\mu _{1}(I_{\alpha }* \vert u \vert ^{ \frac{N+\alpha }{N}}) \vert u \vert ^{\frac{\alpha }{N}-1}u+\beta (I_{\alpha }* \vert v \vert ^{ \frac{N+\alpha }{N}}) \vert u \vert ^{\frac{\alpha }{N}-1}u,\quad x\in {\mathbb{R}}^{N}, \\ -\Delta v+\lambda _{2}(x)v=\mu _{2}(I_{\alpha }* \vert v \vert ^{ \frac{N+\alpha }{N}}) \vert v \vert ^{\frac{\alpha }{N}-1}v+\beta (I_{\alpha }* \vert u \vert ^{ \frac{N+\alpha }{N}}) \vert v \vert ^{\frac{\alpha }{N}-1}v,\quad x\in {\mathbb{R}}^{N}, \\ u, v\in H^{1}({\mathbb{R}}^{N}), \end{cases} $$ { − Δ u + λ 1 ( x ) u = μ 1 ( I α ∗ | u | N + α N ) | u | α N − 1 u + β ( I α ∗ | v | N + α N ) | u | α N − 1 u , x ∈ R N , − Δ v + λ 2 ( x ) v = μ 2 ( I α ∗ | v | N + α N ) | v | α N − 1 v + β ( I α ∗ | u | N + α N ) | v | α N − 1 v , x ∈ R N , u , v ∈ H 1 ( R N ) , where $N\ge 3$ N ≥ 3 , $\mu _{1}, \mu _{2}, \beta >0$ μ 1 , μ 2 , β > 0 , and $\lambda _{1}(x)$ λ 1 ( x ) , $\lambda _{2}(x)$ λ 2 ( x ) are nonnegative functions. The existence of at least one positive ground state of this system is proved under certain assumptions on $\lambda _{1}$ λ 1 , $\lambda _{2}$ λ 2 .