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 .
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