Boundedness and gradient estimates for solutions to Δu + a(x)ulog u + b(x)u = 0 on Riemannian manifolds

2023/09/24 by Wang, Jie, Wang, Youde
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.13686

Abstract

In this paper, combining Nash-Moser iteration and Sallof-Coste type Sobolev ineualities, we establish fundamental and concise C⁰ and C¹ estimates for solutions to a class of nonlinear elliptic equations of the form Δu(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.

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