arXiv2026
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.