Multiple solutions for eigenvalue problems involving an indefinite potential and with (p1(x), p2(x)) balanced growth

In this paper we are concerned with the study of the spectrum for a class of eigenvalue problems driven by two non-homogeneous differential operators with different variable growth and an indefinite potential in the following form

-div[𝒣(x,|∇u|)∇u+𝔌(x,|∇u|)∇u]+V(x)|u|m(x)-2u==λ(|u|q1(x)-2+|u|q2(x)-2)u in Ω,$$\eqalign{ & - {\rm{div}}\left[ {{\cal H}(x,|\nabla u|)\nabla u + \Im (x,|\nabla u|)\nabla u} \right] + V(x)|u{|^{m(x) - 2}}u = \cr & = \lambda \left( {|u{|^{{q_1}(x) - 2}} + |u{|^{{q_2}(x) - 2}}} \right)u\;{\rm{in}}\;\Omega , \cr}$$

which is subjected to Dirichlet boundary condition. The proofs rely on variational arguments and they consist in finding two Rayleigh-type quotients, which lead us to an unbounded continuous spectrum on one side, and the nonexistence of the eigenvalues on the other.