Quasi-linear equation $Δ_pv+av^q=0$ on manifolds with integral bounded Ricci curvature and geometric applications
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| Formato: | Preprint |
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2026
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| author | Wang, Youde Wei, Guodong Zhang, Liqin |
| author_facet | Wang, Youde Wei, Guodong Zhang, Liqin |
| contents | We study nonexistence results and gradient estimates for solutions of \[ Δ_p v + a v^{q}=0 \] defined on complete Riemannian manifolds satisfying a \emph{$χ$-type Sobolev inequality}. We establish a Liouville theorem under the assumptions that the underlying manifold $(M,g)$ supports a \emph{$χ$-type Sobolev inequality} and that the $L^{\fracχ{χ-1}}$-norm of $\ric_-(x)$ is bounded above by a constant depending only on $\dim(M)$, the Sobolev constant $\mathbb{S}_χ(M)$, 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{$χ$-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 $\ric_-(x)\in L^γ$ for some $γ> \fracχ{χ-1}$.
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01837 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quasi-linear equation $Δ_pv+av^q=0$ on manifolds with integral bounded Ricci curvature and geometric applications Wang, Youde Wei, Guodong Zhang, Liqin Differential Geometry Analysis of PDEs We study nonexistence results and gradient estimates for solutions of \[ Δ_p v + a v^{q}=0 \] defined on complete Riemannian manifolds satisfying a \emph{$χ$-type Sobolev inequality}. We establish a Liouville theorem under the assumptions that the underlying manifold $(M,g)$ supports a \emph{$χ$-type Sobolev inequality} and that the $L^{\fracχ{χ-1}}$-norm of $\ric_-(x)$ is bounded above by a constant depending only on $\dim(M)$, the Sobolev constant $\mathbb{S}_χ(M)$, 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{$χ$-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 $\ric_-(x)\in L^γ$ for some $γ> \fracχ{χ-1}$. 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. |
| title | Quasi-linear equation $Δ_pv+av^q=0$ on manifolds with integral bounded Ricci curvature and geometric applications |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2601.01837 |