Uniqueness of solutions for some nonlinear Dirichlet problems

We consider here a class of nonlinear Dirichlet problems, in a bounded domain Ω, of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{ {\begin{array}{*{20}l} { - {\text{div}}(a(x,u)\nabla u) + {\text{div}}(\Phi (u)) = f{\text{ in }}\Omega ,} \\ {u = 0\quad {\text{on }}\partial \Omega ,} \\ \end{array} } \right. $$\end{document}investigating the problem of uniqueness of solutions. The functions Φ(s) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \mapsto a(x,s)$$\end{document} satisfy rather general assumptions of locally Lipschitz continuity (with possibly exponential growth) and the datum f is in L1(Ω). Uniqueness of solutions is proved both for coercive a(x, s) and for the case of a(x, s) degenerating for s large.