= Solution
As printed, the answer is
$$
\boxed{\text{No }q\in[1,\infty]\text{ satisfies the stated estimate for all }u\in H^1(B).}
$$
The constant function $u\equiv1$ 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 is
$$
\boxed{1\leq q\leq6,\qquad
\|u\|_{L^q(B)}\leq C_q\|\nabla u\|_{L^2(B)}\quad(u\in H_0^1(B)).}
$$
This 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.
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