Some classes of eigenvalues problems for generalized p&qp&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&qp&q type eigenvalues problems given by {−div(a(|∇u|p)|∇u|p−2∇u)=λ|u|m−2u,in  Ω,u=0on  ∂Ω, where ΩΩ is a smooth bounded domain in RNRN, N≥3N≥3, 2≤p1m>1 in suitable ranges listed later, aa is a C1C1 real function and λ>0λ>0 is a real parameter. We make 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.