Ground state solutions to the nonlinear Born-Infeld problem

TL;DR

This work establishes the existence of ground state solutions for the nonlinear Born-Infeld equation in $\mathbb{R}^N$ under zero- and positive-mass regimes by a direct variational method that operates on Pohožaev sets, complemented by concentration-compactness/profile decomposition techniques to overcome noncompactness and nonsmoothness. A key innovation is the projection to Pohožaev manifolds $\mathcal{M}_0$ and $\mathcal{M}_1$, yielding ground-state levels $c_0$ and $c_1$ that coincide with mountain-pass levels and provide sharp energy characterizations. The authors derive a Sobolev-type inequality with the best constant tied to the ground-state energy, and prove that minimizers are critical points, with sign properties and radial minimizers established. Furthermore, they construct nonradial solutions for $N\ge 4$ via symmetry reductions, solving an open problem for $N=5$ and generalizing previous results that relied on symmetry. The framework handles the nonsmooth functional setting without resorting to approximations, offering a robust approach for Born-Infeld-type problems and potential extensions to more general nonlinearities $f(x,u)$ or potentials $V(x)$.

Abstract

In the paper we show the existence of ground state solutions to the nonlinear Born-Infeld problem \[ \mathrm{div}\, \left( \frac{\nabla u}{\sqrt{1-|\nabla u|^2}} \right) + f(u) = 0, \quad x \in \mathbb{R}^N \] in the zero and positive mass cases. Moreover, we find a new proof of the Sobolev-type inequality \[ \int_{\mathbb{R}^N} \left(1 - \sqrt{1-|\nabla u|^2}\right) \, dx \geq C_{N,p} \left( \int_{\mathbb{R}^N} |u|^p \, dx \right)^{\frac{N}{N+p}}, \] for $p > 2^*$ as well as the characterization of the optimal constant $C_{N,p}$ in terms of the ground state energy level. Previous approaches relied on approximation schemes and/or symmetry assumptions, which typically yield to compact embeddings and may lead to solutions that are not at the ground state energy level. In contrast, neither approximation arguments nor symmetry assumptions are employed in the paper to obtain a ground state solution. Instead, we develop a new direct variational approach based on minimization over a Pohožaev manifold combined with profile decomposition techniques. Finally, we show that nonradial solutions exist whenever $N \geq 4$; in particular, this settles a previously open problem in the case $N=5$.

Theorems & Definitions (44)

• Remark 1
• Theorem 1.1
• Remark 2
• Theorem 1.2
• Remark 3
• Theorem 1.3
• Theorem 1.4
• Lemma 1
• proof
• Lemma 2