Born-Infeld problem with general nonlinearity

Abstract

In this paper, using variational methods, we look for non-trivial solutions for the following problem $$\begin{cases}-\text{div}\left(a(|\nabla u|^2)\nabla u\right)=g(u), & \text{ in }\mathbb{R}^N, N \ge 3, u(x) \to 0, & \text{ as } |x| \to +\infty\end{cases}$$ 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.