On Nonlocal Choquard System with Hardy–Littlewood–Sobolev Critical Exponents

Standing wave solutions of the following Hartree system with nonlocal interaction and critical exponent are considered: -(a+b∫Ω|∇u|2)Δu=h(x)∫Ω|v(y)|2μ∗|x-y|μdy|u|2μ∗-2u+fλ(x)|u|q-2u,inΩ,-(a+b∫Ω|∇v|2)Δv=h(x)∫Ω|u(y)|2μ∗|x-y|μdy|v|2μ∗-2v+gσ(x)|v|q-2v,inΩ,u,v≥0,inΩ,u,v=0,on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lllll} &{}-(a+b\displaystyle \int _{\Omega }|\nabla u|^2)\Delta u=h(x)\left( \displaystyle \int _{\Omega }\frac{|v(y)|^{2^*_\mu }}{|x-y|^\mu }\mathrm{{d}}y\right) |u|^{2^*_\mu -2}u\\ &{} \qquad \qquad \qquad \qquad \qquad \qquad \qquad + f_\lambda (x)|u|^{q-2}u,\quad in\,\Omega ,\\ &{}-(a+b\displaystyle \int _{\Omega }|\nabla v|^2)\Delta v=h(x)\left( \displaystyle \int _{\Omega }\frac{|u(y)|^{2^*_\mu }}{|x-y|^\mu }\mathrm{{d}}y\right) |v|^{2^*_\mu -2}v\\ &{} \qquad \qquad \qquad \qquad \qquad \qquad \qquad + g_\sigma (x)|v|^{q-2}v,\quad in\,\Omega ,\\ &{} u,v\ge 0, \quad \ in\,\Omega ,\\ &{} u,v=0, \quad \ on\,\partial \Omega , \end{array} \right. \end{aligned}$$\end{document}where 1<q<2,2μ∗=2N-μN-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<q<2, 2^{*}_{\mu }=\frac{2N-\mu }{N-2}$$\end{document} is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. We study the effect of nonlocal interaction on the number of solutions in the case of general response function Ψ(x)=|x|-μ(0<μ<N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi (x)=|x|^{-\mu } (0<\mu <N)$$\end{document}, which possesses more information on the mutual interaction between the particles. When parameters pair (λ,σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\lambda , \sigma )$$\end{document} belongs to a certain subset of R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document}, we prove the existence, nonexistence and the limit behavior of the nonnegative vector solutions depending on parameters. In the special case of q=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=2$$\end{document}, existence of nonnegative solution is also established. Our work extends and develops some recent results in the literature.