Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems. (October 2022) - Garain, Prashanta; Ukhlov, Alexander
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.