A uniform bound for the solutions to a simple nonlinear equation on Riemannian manifolds

Let MM be a complete noncompact manifold with Ricci curvature bounded below. In this note, we derive a uniform bound for the solutions to the nonlinear equation Δu+aulogu=0,on M, where aa is a real constant. Our method is based on the refined global gradient estimates for the corresponding evolution equation, which is due to Yau [S.T. Yau, Harnack inequality for non-self-adjoint evolution equations, Math. Res. Lett. 2 (1995) 387–399]. We partially generalize the result of Yang [Y.Y. Yang, Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds, Proc. Amer. Math. Soc. 136 (2008) 4095–4102]. In the particular case of complete Riemannian manifolds with nonnegative curvature, we get a sharp upper bound for the positive solutions; this upper bound is independent of the dimension of the manifolds.