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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Sharp estimates of radial minimizers of $p$–Laplace equations
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by Miguel Angel Navarro and Salvador Villegas PDF
Proc. Amer. Math. Soc. 145 (2017), 2931-2941 Request permission

Abstract:

We study semi-stable, radially symmetric and decreasing solutions $u\in W^{1,p}(B_1)$ of $-\Delta _p u=g(u)$ in $B_1\setminus \{ 0\}$, where $B_1$ is the unit ball of $\mathbb {R}^N$, $p>1$, $\Delta _p$ is the $p-$Laplace operator and $g$ is a general locally Lipschitz function. We establish sharp pointwise estimates for such solutions, which do not depend on the nonlinearity $g$. By applying these results, sharp pointwise estimates are obtained for the extremal solution and its derivatives (up to order three) of the equation $-\Delta _p u=\lambda f(u)$, posed in $B_1$, with Dirichlet data $u|_{\partial B_1}=0$, where the nonlinearity $f$ is an increasing $C^1$ function with $f(0)>0$ and $\lim _{t\rightarrow +\infty }{\frac {f(t)}{t^{p-1}}}=+\infty .$
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Additional Information
  • Miguel Angel Navarro
  • Affiliation: Departamento de Análisis Matemático, Universidad de Granada, 18071 Granada, Spain
  • Email: mnavarro_2@ugr.es
  • Salvador Villegas
  • Affiliation: Departamento de Análisis Matemático, Universidad de Granada, 18071 Granada, Spain
  • MR Author ID: 365323
  • Email: svillega@ugr.es
  • Received by editor(s): August 4, 2016
  • Published electronically: February 24, 2017
  • Additional Notes: The authors have been supported by the MEC Spanish grant MTM2012-37960
  • Communicated by: Joachim Krieger
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2931-2941
  • MSC (2010): Primary 35B25, 35J92
  • DOI: https://doi.org/10.1090/proc/13454
  • MathSciNet review: 3637942