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1 June 2019 Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient
Marie-Françoise Bidaut-Véron, Marta García-Huidobro, Laurent Véron
Duke Math. J. 168(8): 1487-1537 (1 June 2019). DOI: 10.1215/00127094-2018-0067

Abstract

We study local and global properties of positive solutions of Δu=up|u|q in a domain Ω of RN, in the range p+q>1, p0, 0q<2. We first prove a local Harnack inequality and nonexistence of positive solutions in RN when p(N2)+q(N1)<N. Using a direct Bernstein method, we obtain a first range of values of p and q in which u(x)c(dist(x,Ω))q2p+q1. This holds in particular if p+q<1+4N1. Using an integral Bernstein method, we obtain a wider range of values of p and q in which all the global solutions are constants. Our result contains Gidas and Spruck’s nonexistence result as a particular case. We also study solutions under the form u(x)=rq2p+q1ω(σ). We prove existence, nonexistence, and rigidity of the spherical component ω in some range of values of N, p, and q.

Citation

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Marie-Françoise Bidaut-Véron. Marta García-Huidobro. Laurent Véron. "Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient." Duke Math. J. 168 (8) 1487 - 1537, 1 June 2019. https://doi.org/10.1215/00127094-2018-0067

Information

Received: 26 March 2018; Revised: 6 December 2018; Published: 1 June 2019
First available in Project Euclid: 18 May 2019

zbMATH: 07080117
MathSciNet: MR3959864
Digital Object Identifier: 10.1215/00127094-2018-0067

Subjects:
Primary: 35J62
Secondary: 35B08

Rights: Copyright © 2019 Duke University Press

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Vol.168 • No. 8 • 1 June 2019
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