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

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 $$\mathrm{\Delta }u(x)+a(x)u(x)\ln u(x)+b(x)u(x)=0,$$ 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.