A continuous spectrum for nonhomogeneous differential operators in orlicz-sobolev spaces

Abstract

We study the nonlinear eigenvalue problem - div(a(Vu|)Vu) = λ|u|q(x)- 2 in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in RN with smooth boundary, q is a continuous function, and a is a nonhomogeneous potential. We establish sufficient conditions on a and q such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments. The abstract results of this paper are illustrated by the cases a(t) = tp-2 log(1 + tr) and a(t) = tp-2[log(1 + t)|1.