Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian
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hdl:2117/345174
Tipus de documentArticle
Data publicació2020-01-01
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Abstract
We consider the equation-¿pu=f(u)in a smooth bounded domain ofRn, where¿pis thep-Laplaceoperator. Explicit examples of unbounded stable energy solutions are known ifn=p+4pp-1. Instead, whenn<p+4pp-1, stable solutions have been proved to be bounded only in the radial case or under strong assump-tions onf. In this article we solve a long-standing open problem: we prove an interiorCabound for stablesolutions which holds for every nonnegativef¿C1wheneverp=2and the optimal conditionn<p+4pp-1holds. Whenp¿(1,2), we obtain the same result under the nonsharp assumptionn<5p. These interior esti-mates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when thedomain is strictly convex. Our work extends to thep-Laplacian some of the recent results of Figalli, Ros-Oton,Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutionswhenp=2in the optimal rangen<10
CitacióCabre, X.; Miraglio, P.; Sanchon, M. Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian. "Advances in Calculus of Variations", 1 Gener 2020, p. 1-37.
ISSN1864-8258
Versió de l'editorhttps://www.degruyter.com/document/doi/10.1515/acv-2020-0055/html
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