Ground state solutions to Born-Infeld-Choquard problem

Abstract

In this paper, we investigate the existence and qualitative properties of ground state solutions for the nonlocal Born-Infeld-Choquard problem \begin{equation*} \begin{cases} -{\rm div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+ \omega u=\big(I_\alpha\ast |u|^{p}\big)|u|^{p-2}u, & \hbox{in }\mathbb{R}^N,\; N\geq 3, \\[5mm] u(x)\to 0, &\hbox{as }|x|\to +\infty. \end{cases} \end{equation*} where $p>\frac{N+\alpha}{N}$, $\omega=0,1$ and $0<\alpha<N$. The equation is driven by the mean curvature operator in Lorentz-Minkowski space, motivated by the Born-Infeld nonlinear electromagnetic theory, and is coupled with a Choquard-type nonlocal nonlinearity. Due to the inherent relativistic gradient constraint $|\nabla u| \le 1$, the associated energy functional lacks standard $\mathcal{C}^1$ regularity, preventing the direct use of classical variational techniques. We employ a non-smooth critical point theory on appropriate Poho\v{z}aev-type manifold to establish the existence of ground state solutions. Such non-smooth critical point theorem is abstract and we further show that it can be employed for strongly indefinite problem as well. We also demonstrate that these solutions are radially symmetric, and monotonously decay to zero at infinity.