Optimal $L^p$ Hardy inequalities

Abstract

Let $\mathcal{Q}(φ):=\int_Ω\big(|\nabla φ|^p+V|φ|^p\big)\dnu$ on $\core$, and assume that $\mathcal{Q}\ge 0$. The aim of the paper is to obtain ''as large as possible" nonnegative (optimal) Hardy-type weight $W$ satisfying $$\mathcal{Q}(φ)\geq \int_Ω W|φ|^p\dnu \quad\forall φ\in\core,$$ on punctured domains $\Omega$.

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• Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem