Multiple solutions for eigenvalue problems involving an indefinite potential and with ( p 1 ( x ) , p 2 ( 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 [ H(x, |∇u|)∇u + I(x, |∇u|)∇u ] + V (x)|u|u = = λ ( |u|1 + |u|2 ) u in Ω, 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.

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