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

Abstract

In this paper, we consider the following magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u = \left(\frac{1}{|x|^α}*|u|^{2_α^*}\right) |u|^{2_α^*-2} u + λf(u)\ \textrm{ in }\ \R^N,\] where $2_{\alpha }^{\ast }=\frac{2N-\alpha }{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, $\lambda >0$, $N\ge 3$, $0<\alpha <N$, $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ 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. Under suitable assumptions on different types of nonlinearities $f$, namely, $f(x,u)=\left(\frac{1}{|x{|}^{\alpha }}\ast |u{|}^p\right)|u{|}^{p-2}u$ for $(2N-\alpha )/N<p<2_{\alpha }^{\ast }$, then $f(u)=|u{|}^{p-1}u$ for $1<p<2^{\ast }-1$ and $f(u)=|u{|}^{2^{\ast }-2}u$ (where $2^{\ast }=2N/(N-2)$), 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 $\lambda$.

21 pages