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On critical exponents of a k-Hessian equation in the whole space

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  18 January 2019

Yun Wang  and
Yutian Lei
Yun Wang
Affiliation:
School of Mathematical Sciences, Institute of Mathematics, Nanjing Normal University, Nanjing 210023, China ([email protected])
Yutian Lei
Affiliation:
School of Mathematical Sciences, Institute of Mathematics, Nanjing Normal University, Nanjing 210023, China ([email protected])
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Abstract

In this paper, we study negative classical solutions and stable solutions of the following k-Hessian equation

$$F_k(D^2V) = (-V)^p\quad {\rm in}\;\; R^n$$
with radial structure, where n ⩾ 3, 1 < k < n/2 and p > 1. This equation is related to the extremal functions of the Hessian Sobolev inequality on the whole space. Several critical exponents including the Serrin type, the Sobolev type, and the Joseph-Lundgren type, play key roles in studying existence and decay rates. We believe that these critical exponents still come into play to research k-Hessian equations without radial structure.

Keywords

k-Hessian equationstable solutioncritical exponentLiouville theoremdecay rate

MSC classification

Primary: 35B33: Critical exponents
Secondary: 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1555 - 1575
Copyright
Copyright © Royal Society of Edinburgh 2019 

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