Du Bois-Reymond Type Lemma and Its Application to Dirichlet Problem with the p(t)-Laplacian on a Bounded Time Scale

Jean Mawhin; Ewa Skrzypek; Katarzyna Szymańska-Dȩbowska

doi:10.3390/e23101352

Du Bois-Reymond Type Lemma and Its Application to Dirichlet Problem with the p(t)-Laplacian on a Bounded Time Scale

Entropy (Basel). 2021 Oct 16;23(10):1352. doi: 10.3390/e23101352.

Authors

Jean Mawhin¹, Ewa Skrzypek², Katarzyna Szymańska-Dȩbowska²

Affiliations

¹ Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du Cyclotron, 2, B-1348 Louvain-la-Neuve, Belgium.
² Institute of Mathematics, Lodz University of Technology, Al. Politechniki 10, 93-590 Łódź, Poland.

Abstract

This paper is devoted to study the existence of solutions and their regularity in the p(t)-Laplacian Dirichlet problem on a bounded time scale. First, we prove a lemma of du Bois-Reymond type in time-scale settings. Then, using direct variational methods and the mountain pass methodology, we present several sufficient conditions for the existence of solutions to the Dirichlet problem.

Keywords: direct variational method; du Bois–Reymond lemma; mountain pass lemma; p(t)–Laplacian; time scales; variational methods.