 Solution

As printed, the answer isThe constant function $u\equiv 1$ has zero gradient and positive Lp norm for every such $q$, including its supremum norm when $q=\infty$.

The missing assumption is a zero Sobolev trace, which is present for the weak solutions in the surrounding parts. With $H_0^1(B)$ in place of $H^1(B)$, the intended answer isThis is the three-dimensional Sobolev inequality at $q=6$, followed by Holder inequality on the bounded ball for smaller $q$; it includes the zero-boundary Poincaré inequality at $q=2$. The subsequent parts use this zero-boundary version.