Quasi-linear equation ${\mathrm{\Delta }}_{p}v+a{v}^q=0$ on manifolds with integral bounded Ricci curvature and geometric applications

We study nonexistence results and gradient estimates for solutions of $${\mathrm{\Delta }}_{p}v+a{v}^q=0$$ 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 ${\text{ric}}_{-}(x)$ 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 ${\text{ric}}_{-}(x)\in L^{\gamma }$ 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 $\text{ric}(x)\ge 0$ outside some geodesic ball $B(o,R_0)$. Then there exists a positive constant $C(n)$, depending only on $n$, such that if $$\mathrm{\parallel }{\text{ric}}_{-}{\mathrm{\parallel }}_{L^{\frac{n}{2}}}\le C(n){\mathbb{S}}_{\frac{n}{n-2}}(M),$$ then $(M,g)$ has exactly one end.