Existence and comparison of maximal and minimal solutions for pseudomonotone elliptic problems in \(L^{1}\) (Q1868008)

The paper deals with the Dirichlet problem \[ \begin{cases} -\text{div }(a(x,u,\nabla u))=f& \text{ in } \Omega,\cr u=w & \text{ on } \partial\Omega, \end{cases} \] where \(-\text{div }(a(x,u,\nabla u))\) is a pseudomonotone operator of Leray-Lions type defined in \(W^{1,p}_0(\Omega),\) \(w\in W^{1,p}(\Omega)\) and \(f\in L^1(\Omega).\) Assuming local Lipschitz/Hölder continuity of \(a(x,s,\xi)\) in \(s,\) the authors prove existence of minimal and maximal renormalized solutions as well as comparison results with respect to the data \(f\) and \(w.\) As a particular case, the results include nonmonotone operators of \(p\)-Laplacian type (for any \(p>1\)) for which there is no uniqueness of solutions.