Harnack inequality and principal eigentheory for general infinity Laplacian operators with gradient in $\mathbb{R}^N$ and applications

TL;DR

This paper develops a theory of generalized principal eigenvalues for the infinity Laplacian with gradient, establishing inequalities and applications to maximum principles and positive solutions of Fisher-KPP equations.

Contribution

It introduces new notions of principal eigenvalues for the infinity Laplacian and applies them to analyze maximum principles and solution existence in unbounded domains.

Findings

Established Harnack and boundary Harnack inequalities.

Characterized maximum principle validity via eigenvalues.

Analyzed existence and uniqueness of positive solutions.

Abstract

Under the lack of variational structure and nondegeneracy, we investigate three notions of \textit{generalized principal eigenvalue} for a general infinity Laplacian operator with gradient and homogeneous term. A Harnack inequality and boundary Harnack inequality are proved to support our analysis. This is a continuation of our first work [3] and a contribution in the development of the theory of \textit{generalized principal eigenvalue} beside the works [8, 13, 12, 9, 29]. 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.