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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
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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\vert _{\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
Email: svillega@ugr.es

DOI: https://doi.org/10.1090/proc/13454
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