On a question of Brezis and Marcus
Let \(\Omega\) be a bounded convex domain in \(\mathbb R^n\), \(d(x):=\text{dist}(x,\partial \Omega)\) and let \(D_{\text{int}} := 2 \sup_{x\in\Omega} d(x)\). The aim of the paper is to prove that, for any \(u\in W^{1,p}_0 (\Omega)\), \[ \int_{\Omega}| \nabla u| ^p \,dx - \Big(\frac{p-1}{p}\Big)^p \int_{\Omega} \frac{| u| ^p}{d^p}\, dx \geq cD_{\text{int}}^{-((p(\alpha + n))/q-n+p)} \bigg(\int_{\Omega} d^{\alpha}| u| ^q \,dx\bigg)^{{p}/{q}}, \] with a constant \(c=c(p,q,n,\alpha)>0\) independent of \(\Omega\) and \[ 1<p\leq q \leq \frac{np}{n-p}, \quad n > p, \qquad \alpha > \frac{q}{p}(n-p)-n. \] If \(p=q\), then \(n=p\) is allowed.
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