In the Lieb's paper "On the lowest eigenvalue of the Laplacian for the intersection of two domains" we foundone finds the following remark: Let $u\in L_{loc}^p(\mathbb{R}^N)$ $\nabla u \in L^{p}$ 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 $x$ with $\beta_x$ the 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}
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 thethis inequality \eqref{lb} above is trueholds.
