Ground state solutions to Born-Infeld-Choquard problem

Jaros{\l}aw Mederski; Xiangjian Zeng

arxiv: 2606.15858 · v2 · pith:CMSPCT3Fnew · submitted 2026-06-14 · 🧮 math.AP

Ground state solutions to Born-Infeld-Choquard problem

Jaros{\l}aw Mederski , Xiangjian Zeng This is my paper

Pith reviewed 2026-07-01 07:51 UTC · model grok-4.3

classification 🧮 math.AP

keywords Born-Infeld equationChoquard problemground state solutionsPohožaev manifoldnon-smooth critical pointsmean curvature operatornonlocal nonlinearityradial symmetry

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The pith

Ground state solutions exist for the Born-Infeld-Choquard equation with relativistic gradient constraint by applying non-smooth critical point theory on a Pohožaev-type manifold.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes existence of ground state solutions for a nonlocal equation that combines the mean curvature operator from Born-Infeld theory with a Choquard nonlocal nonlinearity. The relativistic constraint prevents the energy functional from being C¹, so standard variational methods cannot be applied directly. The authors instead work on a Pohožaev-type manifold and invoke an abstract non-smooth critical point theorem to locate the minimizers. They further show that the resulting solutions are radially symmetric and decay monotonically to zero at infinity. The same abstract theorem is shown to handle strongly indefinite problems as well.

Core claim

We prove the existence of ground state solutions to the Born-Infeld-Choquard problem by employing a non-smooth critical point theory on an appropriate Pohožaev-type manifold. The energy functional lacks C¹ regularity due to the constraint |∇u| ≤ 1, but the method still applies. The solutions are radially symmetric and monotonically decay to zero at infinity. The abstract theorem can also be used for strongly indefinite problems.

What carries the argument

Non-smooth critical point theory applied to a Pohožaev-type manifold, which serves as a constraint set to recover mountain-pass geometry despite the energy functional lacking C¹ regularity from the relativistic gradient bound |∇u| ≤ 1.

Load-bearing premise

The abstract non-smooth critical point theorem applies directly to the energy functional on the Pohožaev manifold even though the functional is not C¹ due to the gradient constraint.

What would settle it

A concrete function that minimizes the energy on the Pohožaev manifold yet fails to satisfy the original PDE, or an explicit parameter choice in the given range where no ground state exists.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves existence for ground states of the Born-Infeld-Choquard equation via non-smooth critical point theory on a Pohožaev manifold, a straightforward extension of known methods to a new combination.

read the letter

The main takeaway is that Mederski and Zeng establish existence of ground state solutions for the given nonlocal problem with the relativistic constraint |∇u| ≤ 1. They apply an abstract non-smooth critical point theorem to a Pohožaev-type manifold and note that the same theorem works for strongly indefinite problems. They also obtain radial symmetry and monotonic decay.

What stands out is the specific pairing of the Born-Infeld mean curvature operator with the Choquard term, which does not appear in the prior literature they cite. The adaptation avoids classical C¹ methods because the energy functional is not differentiable under the gradient bound, and they claim the manifold geometry still produces a mountain-pass critical point. That extension to strongly indefinite cases is the part that could interest others working on similar indefinite functionals.

The soft spot is the dependence on the abstract theorem applying without extra verification steps. The lack of C¹ regularity means every hypothesis of the non-smooth result has to hold exactly, including how the Pohožaev constraint interacts with the nonlocal term. The abstract states they check this, but the load-bearing step is whether the functional really satisfies the conditions of the theorem in this setting. No circularity shows up in the description.

This is for specialists in variational methods for nonlocal or relativistic elliptic equations. A reader already following Choquard problems or mean curvature operators in Minkowski space would get a concrete example of the technique. It deserves peer review because the problem statement is new and the argument is presented as a direct application rather than a major new tool.

Referee Report

1 major / 0 minor

Summary. The manuscript proves existence of ground state solutions to the Born-Infeld-Choquard problem driven by the mean curvature operator in Lorentz-Minkowski space coupled with a Choquard nonlocal term, subject to the relativistic constraint |∇u| ≤ 1. The proof adapts an abstract non-smooth critical point theorem to a Pohožaev-type manifold; the solutions are shown to be radially symmetric and monotonically decaying. The paper also presents a general version of the non-smooth theorem applicable to strongly indefinite problems.

Significance. If the verification that the specific energy functional satisfies the hypotheses of the abstract theorem is complete, the result is of interest for extending variational methods to non-C¹ functionals arising from relativistic constraints and nonlocal nonlinearities. The provision of an abstract non-smooth critical point theorem shown to handle strongly indefinite problems is a concrete strength that may be reusable in other settings.

major comments (1)

[the section applying the abstract non-smooth critical point theorem] The section applying the abstract non-smooth critical point theorem: the manuscript must contain an explicit, item-by-item verification that the energy functional satisfies every hypothesis of the abstract theorem (including the required geometry on the Pohožaev manifold and any compactness or Palais-Smale condition), given that the functional is not C¹ because of the constraint |∇u| ≤ 1. Without this verification the central existence claim cannot be confirmed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comment. We agree that an explicit item-by-item verification of the hypotheses is essential for confirming the application of the abstract non-smooth critical point theorem to our non-C¹ functional. We will revise the manuscript to include this verification.

read point-by-point responses

Referee: The section applying the abstract non-smooth critical point theorem: the manuscript must contain an explicit, item-by-item verification that the energy functional satisfies every hypothesis of the abstract theorem (including the required geometry on the Pohožaev manifold and any compactness or Palais-Smale condition), given that the functional is not C¹ because of the constraint |∇u| ≤ 1. Without this verification the central existence claim cannot be confirmed.

Authors: We agree with the referee that the current presentation would benefit from a more explicit, item-by-item check. In the revised manuscript we will insert a new subsection (immediately after the statement of the abstract theorem) that verifies each hypothesis in turn: (i) the geometry of the Pohožaev manifold (including the mountain-pass geometry and the fact that the manifold is a C¹ manifold of codimension one), (ii) the Palais-Smale condition at the mountain-pass level (using the relativistic constraint |∇u|≤1 and the nonlocal term), and (iii) the remaining technical assumptions of the abstract theorem. This verification will be carried out directly on the energy functional restricted to the Pohožaev manifold, taking into account its lack of C¹ regularity. We believe this addition will make the existence proof fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is an existence proof that applies an abstract non-smooth critical point theorem to a Pohožaev-type manifold for a functional that lacks C¹ regularity due to the relativistic constraint |∇u| ≤ 1. The central claim does not reduce to any self-definitional step, fitted input renamed as prediction, or load-bearing self-citation chain; the theorem is invoked as an external abstract tool whose applicability is verified directly on the given functional and manifold geometry. No equations or derivations in the provided text equate the result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The existence claim rests on standard assumptions in variational PDE theory (N ≥ 3, range of p and α) plus the applicability of an abstract non-smooth critical point theorem; no free parameters or invented entities are introduced.

axioms (2)

domain assumption The energy functional satisfies the geometric conditions required by the non-smooth critical point theorem on the Pohožaev manifold.
Invoked to justify the mountain-pass geometry and existence.
domain assumption The relativistic constraint |∇u| ≤ 1 is compatible with the Choquard nonlinearity for the given parameter ranges.
Necessary for the functional to be well-defined.

pith-pipeline@v0.9.1-grok · 5820 in / 1240 out tokens · 19340 ms · 2026-07-01T07:51:50.699947+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

41 extracted references · 3 canonical work pages

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Azzollini,Ground state solution for a problem with mean curvature operator in Minkowski space, J

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2014

[2] [2]

A Azzollini,On a prescribed mean curvature equation in Lorentz-Minkowski space, Journal de Math´ ematiques Pures et Appliqu´ ees,106(2016), 1122–1140. 2

2016

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Baldelli, J

L. Baldelli, J. Mederski, A. Pomponio,Normalized solutions to Born-Infeld and quasilinear problems, Journal of Differential Equations444(2025) 113595 . 2

2025

[4] [4]

Bartnik, L

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1982

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Bartsch, J

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2017

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2012

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2016

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Bonheure, P

D. Bonheure, P. d’Avenia, A. Pomponio, W. Reichel,Equilibrium measures and equilibrium potentials in the Born- Infeld model, J. Math. Pures Appl.139(2020), 35–62. 1

2020

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Bonheure, A

D. Bonheure, A. Iacopetti,On the regularity of the minimizer of the electrostatic Born-Infeld energy, Arch. Ration. Mech. Anal.232(2019), 697–725. 1

2019

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J. Byeon, N. Ikoma, A. Malchiodi, L. Mari,Existence and regularity for prescribed Lorentzian mean curvature hypersurfaces, and the Born–Infeld model,Ann. PDE10, 4 (2024) 2

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J. Byeon, N. Ikoma, A. Malchiodi, L. Mari,Compactness via monotonicity in nonsmooth critical point theory, with application to Born-Infeld type equations.J. Funct. Anal290(11) (2026), 111438. 2, 3, 5, 7, 8, 17, 18

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work page doi:10.1090/tran/9748 2026

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work page arXiv 2006

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Moroz, J

V. Moroz, J. Van Schaftingen,Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal.265(2013), 153–184. 2, 7, 18

2013

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