Bound state positive solutions for a class of elliptic system with Hartree nonlinearity

Abstract

In this paper, we are concerned with the following two-component system of Schrodinger equations with Hartree nonlinearity: \begin{document}$ \begin{equation*} \begin{cases} -\varepsilon ^{2}\Delta u+V_{1}\left( x\right) u+\lambda _{1}\left(I_{\varepsilon }\ast |u|^{p+1}\right)|u|^{p-1}u \qquad\quad\; = \left(\mu _{1}|u|^{2p}+\beta(x) |u|^{q-1}|v|^{q+1}\right)u, & \text{in }\mathbb{R}^{N}, -\varepsilon ^{2}\Delta v+V_{2}\left( x\right) v+\lambda _{2}\left(I_{\varepsilon }\ast |v|^{p+1}\right)|v|^{p-1}v \qquad\quad\; = \left(\mu _{2}|v|^{2p}+\beta(x) |v|^{q-1}|u|^{q+1}\right)v, & \text{in }\mathbb{R}^{N}\,, u,v\in H^{1}(\mathbb{R}^{N}),\quad u,v>0, \end{cases} \end{equation*} $\end{document} where \begin{document}$ 0 is a small parameter, \begin{document}$ 0 , \begin{document}$ I_{\varepsilon}(x) = \frac{\Gamma((N-\alpha)/2)} {\Gamma(\alpha/2)\pi^{\frac{N}{2}}2^{\alpha}\varepsilon^{\alpha}}\frac{1}{|x|^{N-\alpha}}, \; x\in\mathbb{R}^{N}\setminus\{0\} $\end{document} , \begin{document}$ \alpha\in(0,N),\; N = 3,4,5 $\end{document} and \begin{document}$ \lambda _{l}\geq0,\; \mu _{l}>0,\; l = 1,2, $\end{document} are constants. Under some suitable assumptions on the potentials \begin{document}$ V_{l}(x),l = 1,2, $\end{document} and the coupled function \begin{document}$ \beta(x) $\end{document} , we prove the existence and multiplicity of positive solutions for the above system by using energy estimates, the Nehari manifold technique and the Lusternik-Schnirelmann theory. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored.