Dirichlet μ -Parametric Differential Problem with Multivalued Reaction Term

Mina Ghasemi, Calogero Vetro () and Zhenfeng Zhang
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Mina Ghasemi: Department of Mathematics and Computer Sciences, Physical Sciences and Earth Sciences (MIFT), University of Messina, Viale Ferdinando Stagno d’Alcontres, 98166 Messina, Italy
Calogero Vetro: Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy
Zhenfeng Zhang: School of Mathematics, Hohai University, Nanjing 210098, China

Mathematics, 2025, vol. 13, issue 8, 1-15

Abstract: 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.

Keywords: competing operator; differential inclusion; existence of generalized and weak solutions; Galerkin method; ( p , q )-Laplace differential inclusion (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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