Do we have embedding theorem for $\mathcal{E}_{2,p}=\{u\in L^p(\Bbb R^3), \nabla u\in L^2(\Bbb R^3)\}$?

Question

Suppose we have $u\in L^p(\mathbb R^3)$ and $\nabla u\in L^2(\Bbb R^3)$ for some $p\leq 6$, can we conclude that $u\in L^q(\Bbb R^3)$ for all $q\in [p,6]$? If so, how can we show that?

More generally, if $u\in L^p(\mathbb R^3)$ and $\nabla u\in L^q(\Bbb R^3)$ for some suitably chosen $p$ and $q$, do we have some embedding theorem for such $u$?

Any comment or reference will be appreciated! Thanks.