A remark on some nonlinear elliptic problems - Lucio Boccardo (Electronic Journal of Differential Equations, 2002)
We shall prove an existence result of $W_0^{1,p}(Omega)$ solutions for the boundary value problem $$displylines{ -mathopm div a(x, u,abla u)=F quadmboxin Omegacr u=0quadmboxon partialOmega }$$ with right hand side in $W^{-1,p'}(Omega)$. The features of the equation are that no restrictions on the growth of the function $a(x,s,xi)$ with respect to $s$ are assumed and that $a(x,s,xi)$ with respect to $xi$ is monotone, but not strictly monotone. We overcome the difficulty of the uncontrolled growth of $a$ thanks to a suitable definition of solution (similar to the one introduced in citeB6 for the study of the Dirichlet problem in $L^1$) and the difficulty of the not strict monotonicity thanks to a technique (the $L^1$-version of Minty's Lemma) similar to the one used in citeBO.