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
$$
\cases
\displaystyle
-\vr^2\Delta u +V(x)u =\bigg(\int_{\mathbb{R}^N} \frac{|z|^{p}}{|x-y|^{\mu}}dy\bigg)|z|^{p-2}u,
\\
\displaystyle
-\vr^2\Delta v +V(x)v =-\bigg(\int_{\mathbb{R}^N} \frac{|z|^{p}}{|x-y|^{\mu}}dy\bigg)|z|^{p-2}v,
\endcases
$$
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 $\varepsilon\to 0$.