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_\alpha\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.
From: Xiangjian Zeng [view email]
[v1]
Sun, 14 Jun 2026 15:26:15 UTC (26 KB)
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