Some remarks on the regularity of solutions for a class of elliptic equations with measure data (Q2716498)

A nonlinear Dirichlet problem with homogeneous boundary conditions is considered in an open, bounded domain of \(\mathbb{R}^N.\) The equation to be solved is: NEWLINE\[NEWLINE -\text{div} (a(x,u)(1+|u|)^m \nabla u)=\mu \quad \text{in } \Omega,NEWLINE\]NEWLINE with \(\mu \) a bounded Radon measure on \(\Omega.\) The author proves that it is possible to find solutions in \(H^1_0(\Omega)\) if \(m> 1.\) NEWLINENEWLINENEWLINEThen a perturbated equation is considered NEWLINE\[NEWLINE -\text{div} (a(x,u)(1+|u|)^m \nabla u) +(1+|u|)^{r-1} u|\nabla u|^2=\mu \quad \text{in } \OmegaNEWLINE\]NEWLINE and the existence of a weak solution is obtained if \(m> r+1.\)