I got stuck in the following lemma:

**Lemma:** Let $B$ be the unit ball in the 4 dimensional Euclidean space. Suppose that $u\in W^{2,2}(B)$, then $e^{u}\in L^{q}$ for any $q>1$.

As we know this is exactly the critical case of Sobolev's imbedding where we fail to get $L^{\infty}$ bounds. Any suggestion and help would be appreciated.

**My idea of this that fails**:

Note $\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!}\int_{B} |u|^{k}$. By Sobolev imbedding we know that $W^{2,2}(B)\hookrightarrow L^{k}$ for any $k\geq 1$. So we know that

$$\parallel u\parallel_{L^{k}(B)}\leq C_{k} \parallel u \parallel_{W^{2,2}} $$

We can then get :

$\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!} (C_{k})^{k} \parallel u \parallel_{W^{2,2}}^{k}$.

So that it suffices to control $C_{k}$. But based on my calculation, $C_{k} \sim k$, from which it seems this argument would fail.