  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 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.
