Duke Mathematical Journal

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, and Laurent Véron

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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.

Article information

Source
Duke Math. J., Volume 168, Number 8 (2019), 1487-1537.

Dates
Received: 26 March 2018
Revised: 6 December 2018
First available in Project Euclid: 18 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1558145273

Digital Object Identifier
doi:10.1215/00127094-2018-0067

Mathematical Reviews number (MathSciNet)
MR3959864

Zentralblatt MATH identifier
07080117

Subjects
Primary: 35J62: Quasilinear elliptic equations
Secondary: 35B08: Entire solutions

Keywords
elliptic equations Bernstein methods gradient estimates global solutions bifurcations

Citation

Bidaut-Véron, Marie-Françoise; García-Huidobro, Marta; Véron, Laurent. Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient. Duke Math. J. 168 (2019), no. 8, 1487--1537. doi:10.1215/00127094-2018-0067. https://projecteuclid.org/euclid.dmj/1558145273


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