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

Minbo Yang, Yuanhong Wei

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 $\varepsilon\to 0$.

Keywords


Hamiltonian elliptic systems; generalized linking theorem; (PS) condition; variational methods

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