Born-Infeld problem with general nonlinearity

Abstract

In this paper, using variational methods, we look for non-trivial solutions to the following problem{−div(a(|∇u|2)∇u)=g(u),in RN,N≥3,u(x)→0,as |x|→+∞, under general assumptions on the continuous nonlinearity g. We assume growth conditions of g at 0 and, in the zero mass case, 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)→+∞ as s→1−. We find a radial solution to the problem with finite energy.