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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 ) - 2 u = = λ ( | u | q 1 ( x ) - 2 + | u | q 2 ( x ) - 2 ) u     in     Ω , $$\eqalign{ &amp; - {\rm{div}}\left[ {{\cal H}(x,|\nabla u|)\nabla u + \Im (x,|\nabla u|)\nabla u} \right] + V(x)|u{|^{m(x) - 2}}u =  \cr  &amp; = \lambda \left( {|u{|^{{q_1}(x) - 2}} + |u{|^{{q_2}(x) - 2}}} \right)u\;{\rm{in}}\;\Omega , \cr}$$ 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.

analele universităţii "ovidius" constanţa. seria matematică

Multiple solutions for eigenvalue problems involving an indefinite potential and with (p 1(x), p 2(x)) balanced growth