On an elliptic boundary value problem with discontinuous nonlinearity

Abstract

LetΩ⊂Rⁿbe a bounded and Lebesgue-measurable domain with a Lipschitz boundary ∂Ω.Given a mapping f:Ω× R → R, consider the boundary value problem (BVP)

, where F is the superposition operator u→f(·,u(·)),

∂/∂v denoting the outer conormal derivative on ∂Ω.Assume that s→f(x,s)+M s is increasing for some M≤0, that 910 has a weak lower solution and a weak upper solution ū in the Sobolev space Wi[^1,2Ω) such that and Fu is measurable for each , we shall prove by a generalized iteration method that the BWP (1) has the least and the greatest weak solution in . No continuity hypotheses are imposed on the function f.

Keywords:

• Boundary value problems
• elliptic
• weak solutions
• discontinuous nonlinearity

AMS(MOS):

• 35J