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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 $1<q<5$.

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  • $\begingroup$ math.stackexchange.com/questions/1132206/… $\endgroup$ – Will Jagy Feb 3 '15 at 18:36
  • $\begingroup$ @WillJagy Yes I also put this link in my post. $\endgroup$ – JumpJump Feb 3 '15 at 18:55
  • $\begingroup$ Rescale rescaling you can consider the problem, $\min\Vert u\Vert_{H^1_0(B(0,1))}$ given that $\Vert u\Vert_{L^{q+1}}=\rho$. $\endgroup$ – Liviu Nicolaescu Feb 3 '15 at 20:54
  • $\begingroup$ @LiviuNicolaescu Can you provide more details please? Thank you! I also update my post as well. $\endgroup$ – JumpJump Feb 3 '15 at 20:58
  • $\begingroup$ More details: represent $v\in H^1_0(B(0,r))$ as $v(x)=u(x/r)$ for $u\in H^1_0(B(0,1))$, and rephrase conveniently the problem only talking of $ H^1_0(B(0,1))$ and $L^{q+1}(B(0,1))$. $\endgroup$ – Pietro Majer Feb 4 '15 at 1:13

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