Gradient estimates for Δpu + Δqu + a|u|s−1u + b|u|l−1u = 0 on a complete Riemannian manifold and applications

In this paper, we use the Nash–Moser iteration method to study the local and global behaviors of non-negative solutions to the nonlinear elliptic equation [Formula: see text] defined on a complete Riemannian manifold [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are constants and [Formula: see text], with [Formula: see text], is the usual [Formula: see text]-Laplace operator. Under some assumptions on [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], we derive gradient estimates and Liouville-type theorems for non-negative solutions to the above equation. In particular, we show that, if [Formula: see text] is a non-negative entire solution to [Formula: see text] ([Formula: see text]) on a complete non-compact Riemannian manifold [Formula: see text] with non-negative Ricci curvature and [Formula: see text], and [Formula: see text] where [Formula: see text] then [Formula: see text] is a trivial constant solution.