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I have a question about one of my homework question. I have been struggling for a while and I really need some help.

Assume $N>2$ and $u_k$ is a bounded sequence in $W^{1,2}(\mathbb{R}^N)$ satisfying: \begin{equation} \lim_{k\rightarrow\infty}\sup_x\int_{B_1(x)}\lvert u_k(z)\rvert^2dz=0 \end{equation} I need to show $u_k\rightarrow0$ in $L^q$ for any $q\in (2,\frac{2N}{N-2})$.

I could easily show that $u_k\rightarrow_{L^q}0$ holds on any bounded subset of $\mathbb{R}^N$ (using either Holder Inequality or compact embedding of sobolev spaces for bounded sets). But I cannot extend that to $\mathbb{R}^N$.

I will be very thankful if someone can give me some hint.

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  • $\begingroup$ Do you know if the same limit is true for $q$ instead of $2$? $\endgroup$ – Tomás Mar 10 '14 at 16:57
  • $\begingroup$ I am almost sure it's not true for $q=2$. For example, let $\rho$ be a compact support smooth function and define $u_k(x)=k\rho(x/k)$. I think it's easy to verify that the assumptions hold for $u_k$'s but the claim is not true for $q=2$ in this case. $\endgroup$ – Cohlan Mar 11 '14 at 22:52
  • $\begingroup$ Tomás is not asking about replacing $q$ with $2$, he asked about replacing $2$ with $q$ in the formula with $\lim_{k\to\infty} \sup_x$. $\endgroup$ – user127096 Mar 12 '14 at 6:47
  • $\begingroup$ Oh I see I am sorry. I think you are right. Because we can use Holder Inequality, Sobolev Inequality and boundedness of $u_k$'s to see $\lvert\lvert u_k \rvert\rvert_{L^q(B_1)}\leq \lvert\lvert u_k \rvert\rvert^\alpha_{L^2(B_1)}\lvert\lvert u_k \rvert\rvert^{1-\alpha}_{L^{2N/N-2}(B_1)}\leq\lvert\lvert u_k\rvert\rvert^\alpha_{L^2(B_1)} M$ for some $M$ and $\alpha$ $\endgroup$ – Cohlan Mar 12 '14 at 17:23

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