Abstract.
We prove the following formula
$$\int_{\mathbb{R}^{N}} u|u|^{p-2} \Delta u = -(p-1) \int_{\mathbb{R}^{N}}|u|^{p-2}|\nabla u|^{2}$$
for \(u \in {W^{2, p}}(\mathbb{R}^{N})\) 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.
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Metafune, G., Spina, C. An Integration by Parts Formula in Sobolev Spaces. MedJM 5, 357–369 (2008). https://doi.org/10.1007/s00009-008-0155-0
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DOI: https://doi.org/10.1007/s00009-008-0155-0