Existence of solutions for singularly perturbed Hamiltonian elliptic systems with nonlocal nonlinearities

Abstract

In the present paper we study singularly perturbed Hamiltonian elliptic systems with nonlocal nonlinearities $$ \begin{cases} \displaystyle -\varepsilon^2\Delta u +V(x)u =\bigg(\int_{\mathbb{R}^N} \frac{|z|^{p}}{|x-y|^{\mu}}\,dy\bigg)|z|^{p-2}u, \\ \displaystyle -\varepsilon^2\Delta v +V(x)v =-\bigg(\int_{\mathbb{R}^N} \frac{|z|^{p}}{|x-y|^{\mu}}\,dy\bigg)|z|^{p-2}v, \end{cases} $$ where $z=(u,v)\in H^1(\mathbb{R}^N,\mathbb{R}^2)$, $V(x)$ is a continuous real function on $\mathbb{R}^N$, $0< \mu< N$ and $2-{\mu}/{N}< p< (2N-\mu)/(N-2)$. Under suitable assumptions on the potential $V(x)$, we can prove the existence of solutions for small parameter $\varepsilon$ by variational methods. Moreover, if $N> 2$ and $2+(2-\mu)/(N-2)< p< (2N-\mu)/(N-2)$ then the solutions $z_\varepsilon\to 0$ as the parameter $\varepsilon\to 0$.