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.