Quasilinear elliptic problems with critical exponents and discontinuous nonlinearities

Abstract

Using a recent fixed point theorem in ordered Banach spaces by S. Carl and S. Heikkila, we study the existence of weak solutions to nonlinear elliptic problems −diva(x,∇u) = f (x,u) in a bounded domain Ω ⊂ Rn with Dirichlet boundary condition. In particular, we prove that for some suitable function g , which may be discontinuous, and δ small enough, the p -Laplace equation −div(|∇u|p−2∇u) = |u|p−2u+δg(x,u) has a positive solution which goes to 0 as δ → 0+ , where p∗ is the critical exponent. Mathematics subject classification (2010): 35J62, 35D30, 47H10.