Solutions of nonlinear elliptic problems with lower order terms

Abstract

We give an existence result for strongly nonlinear elliptic equations of the form \[ -{\rm div}(a(x,u,\nabla u))+g(x,u,\nabla u)+H(x,\nabla u) = \mu\ \text{in}\ \Omega, \] where the right hand side belongs to $L^1(\Omega)+W^{-1,p'}(\Omega)$ and $- {\rm div}(a(x,u,\nabla u))$ is a Leray--Lions type operator with growth $|\nabla u|^{p-1}$ in $\nabla u$. The critical growth condition on $g$ is with respect to $\nabla u$ and no growth condition with respect to $u$, while the function $H(x,\nabla u)$ grows as $|\nabla u|^{p-1}$.