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

✍ Scribed by Juan Casado-Dı́az; Alessio Porretta

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 247 KB
• Volume: 53
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(02)00299-7

✦ Synopsis

We consider the nonhomogeneous Dirichlet problem -div(a(x; u; ∇u)) = f in ; u = w on @ ;

where -div(a(x; u; ∇u)) is a pseudomonotone operator of Leray-Lions type deÿned in W 1;p 0 ( ), w ∈ W 1;p ( ) and f is in L 1 ( ). Under suitable assumptions of locally Lipschitz, or locally H older, continuity of a(x; s; ) 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.