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 Δ p u + Δ q u + a | u | s − 1 u + b | u | l − 1 u = 0 defined on a complete Riemannian manifold (M , g) , where q > p > 1 , a , b , s , l are constants and Δ z u = div (| ∇ u | z − 2 ∇ u) , with z ∈ { p , q } , is the usual z -Laplace operator. Under some assumptions on p , q , a , b , s and l , we derive gradient estimates and Liouville-type theorems for non-negative solutions to the above equation. In particular, we show that, if u is a non-negative entire solution to Δ p u + Δ q u = 0 (1 < p < q) on a complete non-compact Riemannian manifold M with non-negative Ricci curvature and dim M = n ≥ 3 , and f = | ∇ u | 2 ∈ L β (M) where β > max n (q − p) 2 , 1 + p 2 n n − 2 , 2 n n − 2 , then u is a trivial constant solution. [ABSTRACT FROM AUTHOR]