Local and Global Log-Gradient estimates of solutions to Δₚv+bvq+cvʳ =0 on manifolds and applications

In this paper, we employ the Nash-Moser iteration technique to study local and global properties of positive solutions to the equation Δₚv+bvq+cvʳ =0 on complete Riemannian manifolds with Ricci curvature bounded from below, where b, c∈ R, $p>1$, and q≤ r are some real constants. Assuming certain conditions on $b,\, c,\, p,\, q$ and r, we derive succinct Cheng-Yau type gradient estimates for positive solutions, which is of sharp form. These gradient estimates allow us to obtain some Liouville-type theorems and Harnack inequalities. Our Liouville-type results are novel even in Euclidean spaces. Based on the local gradient estimates and a trick of Sung and Wang, we also obtain the global gradient estimates for such solutions. As applications we show the uniqueness of positive solutions to some generalized Allen-Cahn equation and Fisher-KPP equation.