Born-Infeld problem with general nonlinearity

TL;DR

This paper investigates the existence of finite energy solutions for a class of nonlinear elliptic equations involving a generalized Born-Infeld operator, using variational methods under broad assumptions on the nonlinearity.

Contribution

It introduces a variational framework to find solutions for nonlinear problems with general nonlinearity and a class of operators extending the Born-Infeld model.

Findings

Existence of radial solutions with finite energy.

Applicable to a broad class of nonlinearities without growth restrictions at infinity.

Extension of the classical Born-Infeld problem to more general operators.

Abstract

In this paper, using variational methods, we look for non-trivial solutions for the following problem $$ \begin{cases} -{\rm div}\left(a(|\nabla u|^2)\nabla u\right)=g(u), & \hbox{in }\mathbb{R}^N,\; N\geq 3, \\[1mm] u(x)\to 0, &\hbox{as }|x|\to +\infty, \end{cases} $$ under general assumptions on the continuous nonlinearity $g$. We assume only growth conditions of $g$ at $0$, however no growth conditions at infinity are imposed. If $a(s)=(1-s)^{-1/2}$, we obtain the well-known Born-Infeld operator, but we are able to study also a general class of $a$ such that $a(s)\to+\infty$ as $s\to 1^{-}$. We find a radial solution to the problem with finite energy.