Existence of solutions for singularly perturbed Hamiltonian elliptic systems with nonlocal nonlinearities
Keywords
Hamiltonian elliptic systems, generalized linking theorem, (PS) condition, variational methodsAbstract
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 $\varepsilon\to 0$.Downloads
Published
2016-04-12
How to Cite
1.
YANG, Minbo and WEI, Yuanhong. Existence of solutions for singularly perturbed Hamiltonian elliptic systems with nonlocal nonlinearities. Topological Methods in Nonlinear Analysis. Online. 12 April 2016. Vol. 43, no. 2, pp. 385 - 402. [Accessed 28 March 2024].
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