Nonlinear Perturbations of a periodic magnetic Choquard equation with Hardy-Littlewood-Sobolev critical exponent

Nonlinear Perturbations of a periodic magnetic Choquard equation with Hardy-Littlewood-Sobolev critical exponent

In this paper, we consider the following magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u = \left(\frac{1}{|x|^{\alpha}}*|u|^{2_{\alpha}^*}\right) |u|^{2_{\alpha}^*-2} u + \lambda f(u)\ \textrm{ in }\ \R^N,\] where \(2_{\alpha}^{*}=\frac{2N-\alpha}{N-2}\) is the critical exponent in...

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Veröffentlicht in:arXiv.org 2019-10
Hauptverfasser: Bueno, Hamilton, Lisboa, Narciso, Vieira, Leandro
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Mathematics - Analysis of PDEs
Perturbation
Variational methods
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description In this paper, we consider the following magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u = \left(\frac{1}{|x|^{\alpha}}*|u|^{2_{\alpha}^*}\right) |u|^{2_{\alpha}^*-2} u + \lambda f(u)\ \textrm{ in }\ \R^N,\] where \(2_{\alpha}^{*}=\frac{2N-\alpha}{N-2}\) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, \(\lambda>0\), \(N\geq 3\), \(0
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Under suitable assumptions on different types of nonlinearities \(f\), namely, \(f(x,u)=\left(\frac{1}{|x|^{\alpha}}*|u|^{p}\right)|u|^{p-2} u\) for \((2N-\alpha)/N<p<2^{*}_{\alpha}\), then \(f(u)=|u|^{p-1} u\) for \(1<p<2^*-1\) and \(f(u)=|u|^{2^* - 2}u\) (where \(2^*=2N/(N-2)\)), we prove the existence of at least one ground state solution for this equation by variational methods if \(p\) belongs to some intervals depending on \(N\) and \(\lambda\).]]></abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><doi>10.48550/arxiv.1907.05435</doi><oa>free_for_read</oa></addata></record>
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Variational methods
title Nonlinear Perturbations of a periodic magnetic Choquard equation with Hardy-Littlewood-Sobolev critical exponent
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