Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems. (October 2022)

• Record Type: Journal Article
• Authors: Garain, Prashanta; Ukhlov, Alexander
• Is Part Of: Nonlinear analysis. Volume 223(2022)
• Publication Date: 2022-10
• Subjects: 35R11 -- 35J92 -- 35A23 -- 35J75; Mixed local and nonlocal p-Laplace operator -- Sobolev inequality -- Extremal -- Singular problem -- Existence -- Uniqueness -- Regularity -- Symmetry; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113022
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this article, we consider mixed local and nonlocal Sobolev ( q, p ) -inequalities with extremal in the case 0 < q < 1 < p < ∞ . We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with a singular elliptic problem involving the mixed local and nonlocal p -Laplace operator. Moreover, it is proved that the mixed Sobolev inequalities are necessary and sufficient condition for the existence of weak solutions of such singular problems. As a consequence, a relation between the singular p -Laplace and mixed local and nonlocal p -Laplace equation is established. Finally, we investigate the existence, uniqueness, regularity and symmetry properties of weak solutions for such problems.