Dirichlet μ-Parametric Differential Problem with Multivalued Reaction Term

We study a Dirichlet μ-parametric differential problem driven by a variable competing exponent operator, given by the sum of a negative p-Laplace differential operator and a positive q-Laplace differential operator, with a multivalued reaction term in the sense of a Clarke subdifferential. The parameter μ∈R makes it possible to distinguish between the cases of an elliptic principal operator (μ≤0) and a non-elliptic principal operator (μ>0). We focus on the well-posedness of the problem in variable exponent Sobolev spaces, starting with energy functional analysis. Using a Galerkin approach with a priori estimate and embedding results, we show that the functional associated with the problem is coercive; hence, we prove the existence of generalized and weak solutions.