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The Existence of Solutions for Local Dirichlet ( r ( u ), s ( u ))-Problems

Author

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  • Calogero Vetro

    (Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy)

Abstract

In this paper, we consider local Dirichlet problems driven by the ( r ( u ) , s ( u ) ) -Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r , s are real continuous functions and we have dependence on the solution u . The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument.

Suggested Citation

  • Calogero Vetro, 2022. "The Existence of Solutions for Local Dirichlet ( r ( u ), s ( u ))-Problems," Mathematics, MDPI, vol. 10(2), pages 1-17, January.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:2:p:237-:d:723516
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    Cited by:

    1. Kholoud Saad Albalawi & Mona Bin-Asfour & Francesca Vetro, 2022. "Remarks on Nonlocal Dirichlet Problems," Mathematics, MDPI, vol. 10(9), pages 1-14, May.

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