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Sharp estimates of semistable radial solutions of k-Hessian equations

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  15 March 2019

Miguel Angel Navarro  and
Justino Sánchez
Miguel Angel Navarro
Affiliation:
Departamento de Estatística, Análise Matemática e Optimización Universidade de Santiago de Compostela Santiago de Compostela, 15782, Spain (miguelangel.burgos@usc.es)
Justino Sánchez
Affiliation:
Departamento de Matemáticas, Universidad de La Serena Avenida Cisternas 1200, La Serena, Chile (jsanchez@userena.cl)
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Abstract

We consider semistable, radially symmetric and increasing solutions of Sk(D2u) = g(u) in the unit ball of ℝn, where Sk(D2u) is the k-Hessian operator of u and g ∈ C1 is a general positive nonlinearity. We establish sharp pointwise estimates for such solutions in a proper weighted Sobolev space, which are optimal and do not depend on the specific nonlinearity g. As an application of these results, we obtain pointwise estimates for the extremal solution and its derivatives (up to order three) of the equation Sk(D2u) = λg(u), posed in B1, with Dirichlet data $u\arrowvert _{B_1}=0$, where g is a continuous, positive, nonincreasing function such that lim t→−∞g(t)/|t|k = +∞.

Keywords

k-Hessian operatorsemistable solutionssharp estimatesextremal solution

MSC classification

Primary: 35B35: Stability
Secondary: 35B07: Axially symmetric solutions 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 2083 - 2115
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Copyright
Copyright © Royal Society of Edinburgh 2019

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