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

Calogero Vetro
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Calogero Vetro: Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy

Mathematics, 2022, vol. 10, issue 2, 1-17

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.

Keywords: ( r ( u ), s (u))-Laplacian operator; Palais-Smale condition; monotone operator; regularized problem; weak solution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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