Harnack inequality and principal eigentheory for general infinity Laplacian operators with gradient in RN and applications

Abstract

Under the lack of variational structure and nondegeneracy, we investigate three notions of generalized principal eigenvalue for a general infinity Laplacian operator with gradient and homogeneous term. A Harnack inequality is proved to support our analysis. This is a continuation of our first work (Biswas and Vo in Liouville theorems for infinity Laplacian with gradient and KPP type equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. https://doi.org/10.2422/2036-2145.202105_050) and a contribution in the development of the theory of generalized principal eigenvalue beside the works (Berestycki et al. in Commun Pure Appl Math 47(1):47–92, 1994; Berestycki and Rossi in JEMS 8:195–215, 2006; Berestycki and Rossi in Commun Pure Appl Math 68(6):1014–1065, 2015; Berestycki et al. in J Math Pures Appl 103:1276–1293, 2015; Nguyen and Vo in Calc Var Partial Differ Equ 58(3):102 2019). We use these notions to characterize the validity of maximum principle and study the existence, nonexistence and uniqueness of positive solutions of Fisher-KPP type equations in the whole space. The sliding method is intrinsically improved for infinity Laplacian to solve the problem. The results are related to the Liouville type results, which will be meticulously explained.