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Proceedings of the American Mathematical Society

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Sharp estimates of radial minimizers of $p$–Laplace equations


Authors: Miguel Angel Navarro and Salvador Villegas
Journal: Proc. Amer. Math. Soc. 145 (2017), 2931-2941
MSC (2010): Primary 35B25, 35J92
DOI: https://doi.org/10.1090/proc/13454
Published electronically: February 24, 2017
MathSciNet review: 3637942
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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
Article copyright: © Copyright 2017 American Mathematical Society