Liouville theorems and new gradient estimates for positive solutions to $Δ_pv+a(v+b)^q=0$ on a complete manifold
Youde Wang, Linqin Zhang
TL;DR
The paper studies positive solutions to the nonlinear elliptic equation $Δ_p v + a (v+b)^q = 0$ on complete manifolds with a Ricci lower bound, deriving new gradient estimates of Cheng–Yau type via Saloff-Coste Sobolev inequalities and Nash–Moser iteration. By constructing the auxiliary function $F = f/u^\beta$ with $f = |\nabla u|^2$ and $u = v+b$, the authors obtain pointwise $L_p$-linearizations and crucial integral inequalities that feed into an iterative scheme to produce $L^{\infty}$ bounds on the gradient. The results cover a range of $(a,p,q)$ and provide explicit gradient bounds of the form $\sup_{B(x_0,R/2)} \frac{|\nabla v|^2}{(v+b)^\beta} \le C \frac{1+\kappa R^2}{R^2} \phi_\beta^{2-\beta}$, with implications including relations between log-gradient (Cheng–Yau) estimates and Harnack inequalities, as well as Liouville-type nonexistence results in several regimes. These findings extend known results for the case $p=2$ to general $p>1$ and enrich the understanding of nonlinear elliptic equations on manifolds, providing tools for further geometric-analytic applications.
Abstract
In this paper, we use the Saloff-Coste Sobolev inequality and Nash-Moser iteration method to study the local and global behaviors of positive solutions to the nonlinear elliptic equation $Δ_pv+a(v+b)^q=0$ defined on a complete Riemannian manifold $\left(M,g\right)$ with Ricci lower bound, where $p>1$ is a constant and $Δ_pv=\mathrm{div}\left(\left|\nabla v\right|^{p-2}\nabla v\right)$ is the usual $p$-Laplace operator. Under certain assumptions on $a$, $p$ and $q$, we derive some gradient estimates and Liouville type theorems for positive solutions to the above equation. In particular, under certain assumptions on $a$, $b$, $p$ and $q$ we show whether or not the exact Cheng-Yau $\log$-gradient estimates for the positive solutions to $Δ_pv+av^q=0$ on $\left(M,g\right)$ with Ricci lower bound hold true is equivalent to whether or not the positive solutions to this equation fulfill Harnack inequality, and hence some new Cheng-Yau $\log$-gradient estimates are established.