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.