Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

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.