On the Generalization of the Courant Nodal Domain Theorem

Abstract

In this paper we consider the analogue of the Courant nodal domain theorem for the nonlinear eigenvalue problem for the p-Laplacian. In particular we prove that if uλn is an eigenfunction associated with the nth variational eigenvalue, λn, then uλn has at most 2n−2 nodal domains. Also, if uλn has n+k nodal domains, then there is another eigenfunction with at most n−k nodal domains.