Existence of solutions for singularly perturbed Hamiltonian elliptic systems with nonlocal nonlinearities
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Hamiltonian elliptic systems, generalized linking theorem, (PS) condition, variational methodsAbstrakt
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 $\varepsilon\to 0$.Pobrania
Opublikowane
2016-04-12
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1.
YANG, Minbo & WEI, Yuanhong. Existence of solutions for singularly perturbed Hamiltonian elliptic systems with nonlocal nonlinearities. Topological Methods in Nonlinear Analysis [online]. 12 kwiecień 2016, T. 43, nr 2, s. 385–402. [udostępniono 22.7.2024].
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