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

Abstract: We consider nonexistence and gradient estimate for solutions to $Δpv +av^{q}=0$ 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.