Let $B(0,r)\subset \mathbb R^3$ be a ball centered at $0$ with radius $r$. Define $$ \mathcal A_r:=\{u\in H_0^1(B(0,r)),\,\,\|u\|_{L^{q+1}}=1\}$$ where $1<q<5$. Hence we know that each $\mathcal A_r$ is weakly closed with respect to $H_0^1$ norm.

Now I am interesting in to see whether I can find a constant $C>0$ such that $$ \max_{r>0}\min_{u\in\mathcal A_r}\int_{B(0,r)}|\nabla u|^2\leq C $$

Could this be possible? If no, please provide me a conterexample.

I also put this question in MSE here, please feel free to ask me to delete it if you feel it is duplicated. Thank you.

Update:

It looks to me that if $1+\epsilon<q<5$ will work, but not just $1<q<5$.