An Integration by Parts Formula in Sobolev Spaces

We prove the following formula \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int_{\mathbb{R}^{N}} u|u|^{p-2} \Delta u = -(p-1) \int_{\mathbb{R}^{N}}|u|^{p-2}|\nabla u|^{2}$$\end{document}for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u \in {W^{2, p}}(\mathbb{R}^{N})$$\end{document} 1 < p < + ∞, and related more general results. The equality above easily follows by integrating by parts for p ≥ 2. The case 1  <  p  <  2 is more involved because of the presence of the singularity of |u|p-2 near the zeroes of u and a sectional characterization of Sobolev spaces is required.