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.