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

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