Some classes of eigenvalues problems for generalized p&q-Laplacian type operators on bounded domains

The aim of this paper is to present a preliminary study on a more general class of p&q type eigenvalues problems given by {-div(a(vertical bar del u vertical bar(p))vertical bar del u vertical bar(p-2)del u) = lambda vertical bar u vertical bar(m-2)u, in Omega, u = 0 on partial derivative Omega, where Omega is a smooth bounded domain in R-N, N >= 3, 2 <= p < N, m is an element of R with m > 1 in suitable ranges listed later, a is a C-1 real function and lambda > 0 is a real parameter. Wemake use of variational approaches like a Generalized version of Weierstrass Theorem and/or the Ekeland variational principle and the Mountain Pass Theorem due to Ambrosetti-Rabinowitz. In particular the existence of a continuous and unbounded set of positive generalized eigenvalues is established. (C) 2014 Elsevier Ltd. All rights reserved.