Quasi-linear equation $Δ_pv+av^q=0$ on manifolds with integral bounded Ricci curvature and geometric applications

TL;DR

The paper studies the quasi-linear elliptic equation $\Delta_p v + a v^q = 0$ on complete Riemannian manifolds that support a $\chi$-type Sobolev inequality, establishing Liouville-type nonexistence and gradient estimates under integral Ricci curvature bounds. It develops a p-Laplacian framework via a logarithmic transform, derives precise differential inequalities for the gradient quantity $f=|\nabla u|^2$ with $u=-(p-1)\log v$, and uses Nash–Moser iteration to obtain $L^{\theta\chi}$ gradient bounds and local gradient estimates under $\mathrm{Ric}_- \in L^{\gamma}$. A parallel treatment of the Laplace case ($p=2$) with two dimension-dependent subcases yields Liouville-type nonexistence via auxiliary functions, accompanied by integral estimates and volume-growth considerations. Geometric consequences include lower bounds on ball volumes, uniqueness of the end under small $L^{n/2}$ Ricci bounds relative to the Sobolev constant, and gap-type results for the ends, improving previously known results under weaker hypotheses. The methods blend linearization, sharp integral inequalities, and Nash–Moser iteration to extend Liouville-type phenomena to manifolds with integral Ricci control and to deduce global geometric structure from analytic information.

Abstract

We consider nonexistence and gradient estimate for solutions to $Δ_pv +av^{q}=0$ defined on a complete Riemannian manifold with {\it $χ$-type Sobolev inequality}. A Liouville theorem on this equation is established if the lying manifold $(M, g)$ supports a {\it $χ$-type Sobolev inequality} and the $L^{\fracχ{χ-1}}$ norm of $\ric_-(x)$ of $(M, g)$ is bounded from upper by some constant depending on $\dim(M)$, Sobolev constant $\mathbb{S}_χ(M)$ and volume growth order of geodesic ball $B_r\subset M$. This extends and improves some conclusions obtained recently by Ciraolo, Farina and Polvara \cite{CFP}, but our method employed in this paper is different from their ``P-function" method. In particular, for such manifold with a {\it $χ$-type Sobolev inequality}, we give the lower estimate of volume growth of geodesic ball. If $χ\leq n/(n-2)$, we also establish the local logarithm gradient estimate for positive solutions to this equation under the condition $\ric_-(x)$ is $L^γ$-integrable where $γ>{\fracχ{χ-1}}$. As topological applications of main results(see \corref{main5}) we show that for a complete noncompact Riemannian manifold on which the Sobolev inequality \eqref{chi-n} holds true, $\dim(M)=n\geq 3$ and $\ric(x)\geq 0$ outside some geodesic ball $B(o,R_0)$, there exists a positive constant $C(n)$ depending only on $n$ such that, if $$\|\ric_-\|_{L^{\frac{n}{2}}}\leq C(n)\mathbb{S}_{\frac{n}{n-2}}(M),$$ then $(M, g)$ is of a unique end.

Theorems & Definitions (42)

• Definition
• Theorem : Theorem 1.5 in CFP
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Remark 1
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7