Boundedness and gradient estimates for solutions to $\Delta u + a(x)u\log u + b(x)u = 0$ on Riemannian manifolds

TL;DR

This paper derives fundamental $C^0$ and $C^1$ estimates for solutions to a nonlinear elliptic PDE on Riemannian manifolds, using Nash-Moser iteration and Sobolev inequalities, leading to geometric insights and a Liouville theorem.

Contribution

It introduces new gradient and boundedness estimates for a class of nonlinear elliptic equations on manifolds, combining advanced iterative and Sobolev techniques.

Findings

Established $C^0$ and $C^1$ estimates for solutions.

Derived a local Liouville type theorem.

Connected solutions' properties to geometric structures.

Abstract

In this paper, combining Nash-Moser iteration and Sallof-Coste type Sobolev ineualities, we establish fundamental and concise $C^0$ and $C^1$ estimates for solutions to a class of nonlinear elliptic equations of the form $$\Delta u(x)+a(x)u(x)\ln u(x)+b(x)u(x)=0,$$ which possesses abundant geometric backgrounds. Utilizing these estimates which retrieve more geometric information, we obtain some further properties of such solutions. Especially, we prove a local Liouville type theorem of corresponding constant coefficient equation.