A combined variational–monotone iterative method for elliptic boundary value problems with discontinuous nonlinearity

Abstract

By combining the method of upper and lower solution and monotone iteration with variational in¬equality techniques a constructive existence result for elliptic BVPs with discontinuous nonlinearity is proved. The discontinuous nonlinearity f: R → R is supposed to admit a decomposition of the form f = g − h with functions g,h: R → R that are non-decreasing and in general discontinuous as well. Thus we are able to deal with nonlinearities f that may have jumps in both directions upward and downward which generalizes recent results e.g. of Ambrosetti and Turner [2], Ambrosetti and Badiale [3], Heikkila [17] and the author [8]

Keywords:

• Elliptic boundary value problem
• discontinuous nonlinearities
• variational inequalities

AMS(MOS)::

• 35J20
• 35R05
• 49A29