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

TL;DR

This paper investigates nonexistence results, gradient estimates, and geometric properties of solutions to a quasi-linear PDE on manifolds with integral Ricci curvature bounds, extending Liouville theorems and analyzing topological implications.

Contribution

It establishes new Liouville theorems and gradient estimates for solutions on manifolds with integral Ricci bounds, using a novel approach different from previous methods.

Findings

Liouville theorem under Sobolev inequality and Ricci bounds

Lower volume growth bounds for geodesic balls

Uniqueness of ends under Ricci curvature conditions

Abstract

We study nonexistence results and gradient estimates for solutions of \[ \Delta_p v + a v^{q}=0 \] defined on complete Riemannian manifolds satisfying a \emph{$\chi$-type Sobolev inequality}. We establish a Liouville theorem under the assumptions that the underlying manifold $(M,g)$ supports a \emph{$\chi$-type Sobolev inequality} and that the $L^{\frac{\chi}{\chi-1}}$-norm of $\ric_-(x)$ is bounded above by a constant depending only on $\dim(M)$, the Sobolev constant $\mathbb{S}_\chi(M)$, and the volume growth rate of geodesic balls $B_r\subset M$. This extends and improves several recent results of Ciraolo, Farina, and Polvara \cite{CFP}; our approach, however, differs from their ``$P$-function'' method. In addition, for manifolds satisfying a \emph{$\chi$-type Sobolev inequality}, we obtain a lower bound on the volume growth of geodesic balls. We also derive a local logarithmic gradient estimate for positive solutions, assuming that $\ric_-(x)\in L^\gamma$ for some $\gamma > \frac{\chi}{\chi-1}$. Some geometric and topological applications of our main result are also presented in this article (see \thmref{end}, \thmref{main4}, and \corref{main5}). In particular, we prove the following. Let $(M,g)$ be a complete noncompact Riemannian manifold of dimension $n\ge 3$ on which the Sobolev inequality \eqref{chi-n} holds, and assume that $\ric(x)\ge 0$ outside some geodesic ball $B(o,R_0)$. Then 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 $(M,g)$ has exactly one end.