In Lieb's paper "On the lowest eigenvalue of the Laplacian
for the intersection of two domains" one finds the following remark:

> Let $u\in L_{loc}^p(\mathbb{R}^N)$ with $\nabla u \in L^{p}$ and $\|\nabla u\|_{p} \leq 1 .$ Set $k=1+\|u\|_{p}^{-p}\left(\text { for }\|u\|_{p} \leq \infty\right)$. Let $B_{x}$ denote the unitary ball in
 $\mathbb{R}^N$ centered at $x$, and let $\beta_x$ be its characteristic function. Clearly there is some $x$ such that
\begin{equation}
\label{lb}
\int|\nabla u|^{p} \beta_{x}<k \int|u|^{p} \beta_{x}.
\end{equation}

I can't see why this inequality holds.