## Topological Methods in Nonlinear Analysis

### 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 $\varepsilon\to 0$.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 43, Number 2 (2014), 385-402.

Dates
First available in Project Euclid: 11 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1460381514

Mathematical Reviews number (MathSciNet)
MR3236976

Zentralblatt MATH identifier
1362.35135

#### Citation

Yang, Minbo; Wei, Yuanhong. Existence of solutions for singularly perturbed Hamiltonian elliptic systems with nonlocal nonlinearities. Topol. Methods Nonlinear Anal. 43 (2014), no. 2, 385--402. https://projecteuclid.org/euclid.tmna/1460381514

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