Existence and comparison of maximal and minimal solutions for pseudomonotone elliptic problems in L1 Original Research Article

We consider the nonhomogeneous Dirichlet problem View the MathML source where −div(a(x,u,∇u)) is a pseudomonotone operator of Leray–Lions type defined in View the MathML source, w∈W1,p(Ω) and f is in L1(Ω). Under suitable assumptions of locally Lipschitz, or locally Hölder, continuity of a(x,s,ξ) with respect to s, we prove the existence of maximal and minimal renormalized solutions and comparison results with respect to data f and w. The results include examples of nonmonotone operators of p–laplace type (for any p>1), for which it is known that uniqueness of solutions does not hold.