Ground state solutions to the nonlinear Born-Infeld problem

TL;DR

This paper proves the existence of ground state solutions for the nonlinear Born-Infeld equation in various cases using a novel variational method, and establishes related Sobolev inequalities without relying on symmetry or approximation.

Contribution

It introduces a new direct variational approach on a Pohožaev manifold to find ground states without symmetry assumptions or approximation schemes.

Findings

Existence of ground state solutions in zero and positive mass cases

New proof of a Sobolev-type inequality for the Born-Infeld problem

Nonradial solutions exist for dimensions N ≥ 4

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 Poho\v{z}aev 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 $N=5$.