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

• Autor(a) principal: Leandro da Luz Vieira
• Data de Publicação: 2020
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: http://hdl.handle.net/1843/35582
• topic: Variational methods; Magnetic Choquard equation; Fractional magnetic Choquard equation; Hardy-Littlewood-Sobolev critical exponent; Matemática – Teses; Métodos variacionais – Teses; Equação de Choquard – Teses; Expoente crítico de Hardy­ Littlewood­Sobolev– Teses.

In this work we consider the following magnetic nonlinear Choquard equations \[-(\nabla+iA(x))^2u+ V(x)u = \left(\frac{1}{|x|^{\alpha}}*|u|^{2_{\alpha}^*}\right) |u|^{2_{\alpha}^*-2} u + \lambda f(u)\ \textrm{ in }\ \R^N (N\geq 3)\] and \[(-\Delta)^s_A u+ V(x)u = \left(\frac{1}{|x|^{\alpha}}*|u|^{2_{\alpha,s}^*}\right) |u|^{2_{\alpha,s}^*-2} u + \lambda g(u)\ \textrm{ in }\ \R^N (N=3),\] where $s\in(0,1)$, $2_{\alpha}^{*}=\frac{2N-\alpha}{N-2}$ and $2_{\alpha,s}^{*}=\frac{6-\alpha}{3-2s}$ are critical exponents in the sense of the Hardy-Littlewood-Sobolev inequality. Moreover, in both problems $0<\alpha< N,$ $\lambda>0,$ $A: \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is an $C^1$, $\mathbb{Z}^N$-periodic vector potential and $V$ is a continuous scalar potential given as a perturbation of a periodic potential. Considering different types of nonlinearities $f$ and $g$, 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 $2^*=2N/(N-2)$), $g(x,u)=\left(\frac{1}{|x|^{\alpha}}*|u|^{p}\right)|u|^{p-2} u$ for $(6-\alpha)/3<p<2^{*}_{\alpha,s}$, then $g(u)=|u|^{p-1} u$ for $1<p<2_s^*-1$ and $g(u)=|u|^{2_s^* - 2}u$ (where $2_s^*=6/(3-2s)$), we prove the existence of at least one ground state solution for these equations by variational methods if $p$ belongs to some intervals depending on $N$, $\lambda$ and also on $s$ in the second problem.