Duke Mathematical Journal
- Duke Math. J.
- Volume 168, Number 8 (2019), 1487-1537.
Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient
We study local and global properties of positive solutions of in a domain of , in the range , , . We first prove a local Harnack inequality and nonexistence of positive solutions in when . Using a direct Bernstein method, we obtain a first range of values of and in which . This holds in particular if . Using an integral Bernstein method, we obtain a wider range of values of and 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 . We prove existence, nonexistence, and rigidity of the spherical component in some range of values of , , and .
Duke Math. J., Volume 168, Number 8 (2019), 1487-1537.
Received: 26 March 2018
Revised: 6 December 2018
First available in Project Euclid: 18 May 2019
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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