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

Jie Wang; Youde Wang

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arXiv · 2309.13686

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

Jie Wang·Youde Wang

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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.

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BibTeX RIS

Jie Wang, Youde Wang. 2023-09-24. Boundedness and gradient estimates for solutions to $\Delta u + a(x)u\log u + b(x)u = 0$ on Riemannian manifolds. https://arxiv.org/abs/2309.13686

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