Ground state solutions to Born-Infeld-Choquard problem

Technical Summary: Ground State Solutions to the Born-Infeld-Choquard Problem

Problem Statement
This paper investigates the existence and qualitative properties of ground state solutions (nontrivial least energy solutions) for the nonlocal Born-Infeld-Choquard equation in $\mathbb{R}^N$ ($N \geq 3$):
$$\begin{cases} -\text{div}\left(\frac{\nabla u}{\sqrt{1 - |\nabla u|^2}}\right) + \omega u = (I_\alpha * |u|^p)|u|^{p-2}u, & \text{in } \mathbb{R}^N, \\ u(x) \to 0, & \text{as } |x| \to +\infty, \end{cases}$$
where $0 < \alpha < N$, $p > \frac{N+\alpha}{N}$, and $\omega \in \{0, 1\}$. The operator on the left-hand side is the mean curvature operator in Lorentz-Minkowski space, arising from Born-Infeld nonlinear electrodynamics, which imposes the physical constraint $|\nabla u| \leq 1$. The right-hand side represents a Choquard-type nonlocal nonlinearity involving the Riesz potential $I_\alpha$. The study covers two cases: the positive-mass case ($\omega = 1$) and the zero-mass case ($\omega = 0$).

Methodological Challenges and Approach
The primary difficulty in analyzing this problem stems from the Born-Infeld operator. The associated energy functional is not of class $C^1$ on its natural domain due to the gradient constraint $|\nabla u| \leq 1$ and the singular behavior of the integrand as $|\nabla u| \to 1$. Consequently, classical variational techniques (such as the standard Mountain Pass Theorem) cannot be applied directly.

To overcome this, the authors employ a non-smooth critical point theory framework (based on the work of Szulkin). The methodology involves:

1. Functional Setting: Defining a reflexive Banach space $X_\omega$ and an effective domain $D(\Psi_0)$ where the gradient constraint holds. The energy functional is decomposed into a convex, lower semicontinuous part $\Psi$ (the Born-Infeld term) and a $C^1$ part $\Phi$ (the mass and nonlocal terms).
2. Pohožaev Manifold: Constructing a Pohožaev-type manifold $\mathcal{M}_\omega = \{u \in D(\Psi_0) \setminus \{0\} : P_\omega(u) = 0\}$, where $P_\omega$ is the Pohožaev functional derived from the scaling properties of the equation. The ground state level $c_\omega$ is defined as the infimum of the energy functional on this manifold.
3. Concentration-Compactness: Utilizing a profile decomposition adapted to the nonlocal setting and the specific constraints of the Born-Infeld operator. This includes proving a nonlocal Brezis-Lieb type lemma and a vanishing lemma to handle the lack of compactness in $\mathbb{R}^N$.
4. Abstract Critical Point Theorem: Proving that a minimizer on the Pohožaev manifold corresponds to a critical point of the non-smooth functional in the convex sense, which implies a weak solution to the equation.

Key Contributions and Results

• Existence of Ground States (Positive Mass, $\omega=1$):
  The authors prove that for $p > \frac{N+\alpha}{N}$, the infimum $c_1$ is achieved by a function $u_1 \in \mathcal{M}_1$. This minimizer is a classical ground state solution. The proof relies on showing that minimizing sequences are bounded and, up to translations, converge strongly to a nontrivial limit.

• Existence of Ground States and Sobolev Inequality (Zero Mass, $\omega=0$):
  For the zero-mass case, the existence of a ground state is established for $p > \frac{N+\alpha}{N-2}$. Furthermore, the authors characterize the best constant $C_{p,\alpha}$ for a nonlocal Sobolev-type inequality:
  $$\int_{\mathbb{R}^N} \left(1 - \sqrt{1 - |\nabla u|^2}\right) dx \geq C_{p,\alpha} \left( \int_{\mathbb{R}^N} (I_\alpha * |u|^p)|u|^p dx \right)^{\frac{N}{N+\alpha+2p}}.$$
  They show that the minimizers of this inequality are precisely the ground state solutions of the equation with $\omega=0$.

• Qualitative Properties:
  The paper establishes that any ground state solution $u$ possesses the following properties:

  • Constant Sign: $u$ is either strictly positive or strictly negative on $\mathbb{R}^N$.
  • Radial Symmetry: $u$ is radially symmetric about some point $x_0 \in \mathbb{R}^N$.
  • Monotonic Decay: $u$ decays monotonically to zero as $|x| \to \infty$.
    These properties are derived using polarization techniques and the strong maximum principle, adapted to the non-smooth and nonlocal context.

• Non-Existence in Critical Cases:
  The authors demonstrate the non-existence of nontrivial weak solutions in the critical cases where $p = \frac{N+\alpha}{N}$ (for $\omega=1$) and $p = \frac{N+\alpha}{N-2}$ (for $\omega=0$), provided the gradient term satisfies specific integrability conditions.

Significance and Claims
The paper claims to resolve the existence of ground states for the Born-Infeld-Choquard problem, a case that had remained open due to the combination of the relativistic gradient constraint and nonlocal nonlinearity. The authors emphasize that their approach extends non-smooth critical point theory to strongly indefinite problems and nonlocal settings.

Specifically, the paper highlights that:

1. The developed abstract lemma (Lemma A.1) regarding constrained minimizers on suitable sets is applicable not only to the Born-Infeld-Choquard problem but also to strongly indefinite Schrödinger equations in the settings of Szulkin and Weth.
2. The results provide the first proof of radial symmetry and monotonic decay for ground states in the local case ($\alpha=0$) of the Born-Infeld equation, a property not previously established in the literature (specifically referencing [7]).
3. The work bridges the gap between local Born-Infeld problems and nonlocal Choquard problems, offering a unified variational framework for equations driven by the mean curvature operator in Minkowski space.

The authors do not propose experimental applications or future implications beyond the mathematical scope of establishing existence, regularity, and symmetry for these specific partial differential equations.