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Some results for semi-stable radial solutions of k-Hessian equations with weight on ℝn

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  15 November 2022

Miguel Angel Navarro Open the ORCID record for Miguel Angel Navarro [Opens in a new window]  and
Justino Sánchez Open the ORCID record for Justino Sánchez [Opens in a new window]
Miguel Angel Navarro
Affiliation:
Departamento de Matemáticas, Universidade da Coruña, Campus de Esteiro, Rúa Mendizábal s/n, 15403 Ferrol, A Coruña, Spain (miguel.navarro.burgos@gmail.com)
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 devote this paper to study semi-stable nonconstant radial solutions of $S_k(D^2u)=w(\left \vert x \right \vert )g(u)$ on the Euclidean space $\mathbb {R}^n$. We establish pointwise estimates and necessary conditions for the existence of such solutions (not necessarily bounded) for this equation. For bounded solutions we estimate their asymptotic behaviour at infinity. All the estimates are given in terms of the spatial dimension $n$, the values of $k$ and the behaviour at infinity of the growth rate function of $w$.

Keywords

k-Hessian operatorsemi-stable solutionsweight functions

MSC classification

Primary: 35B35: Stability
Secondary: 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 5 , October 2023 , pp. 1751 - 1776
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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