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(\dfrac{\n u}{\sqrt{1-|\n u|^2}}\right)+ u=\big(I_α\ast |u|^{p}\big)|u|^{p-2}u, & \hbox{in }\RN,\; N\geq 3, \\[5mm] u(x)\to 0, &\hbox{as }|x|\to +\infty. \end{cases} \end{equation*} where $p>\frac{N+\a}{N}$ and $0<\a<N$. 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 Pohožaev-type manifold to establish the existence of ground state solutions. We further demonstrate that these solutions are radially symmetric, and monotonously decay to zero at infinity.