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Sharp estimates of semistable radial solutions of k-Hessian equations

Published online by Cambridge University Press:  15 March 2019

Miguel Angel Navarro
Affiliation:
Departamento de Estatística, Análise Matemática e Optimización Universidade de Santiago de Compostela Santiago de Compostela, 15782, Spain ([email protected])
Justino Sánchez
Affiliation:
Departamento de Matemáticas, Universidad de La Serena Avenida Cisternas 1200, La Serena, Chile ([email protected])

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 gC1 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 = +∞.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

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