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In this article, we consider mixed local and nonlocal Sobolev$(q,p)$-inequalities with extremal in the case $0<q<1<p<\infty$. We prove thatthe extremal of such inequalities is unique up to a multiplicative constantthat is associated with a singular elliptic problem involving the mixed localand nonlocal $p$-Laplace operator. Moreover, it is proved that the mixedSobolev inequalities are necessary and sufficient condition for the existenceof weak solutions of such singular problems. As a consequence, a relationbetween the singular $p$-Laplace and mixed local and nonlocal $p$-Laplaceequation is established. Finally, we investigate the existence, uniqueness,regularity and symmetry properties of weak solutions for such problems.

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