Sobolev Embedding and Uniform $C^1$ bound

Question

I am currently reading a paper and am a little confused about the following, which for clarity, I distill into the following question:

Suppose $\{w_i\} \subset C^2(\mathbb{R}^n)$ is a sequence of real-valued functions, which satisfy for every $R > 0$,

• $-\Delta w_i = e^{w_i}$ on $B_R$, and
• $\|w_i\|_{C^1(B_R)} \leq C$, where $C$ is a positive constant independent of $i$.

Is it true that we can find a subsequence converging weakly in $W^{2,p}(B_R)$, where $p > n$? (I believe that is the claim.)

My limited knowledge of Sobolev embeddings doesn't yield any relation between uniform $C^1$ bounds and $W^{2,p}$ spaces. Is it in this case related to the PDE each $w_i$ satisfies?

Answer 1

You can apply the $L^p$ estimates to get bounds in the $W^{2,p}$ space, take a look at section 9.5 of Gilbarg and Trudinger's book.