Positive Solutions of $p$-th Yamabe Type Equations on Graphs

Abstract

Let $G=(V,E)$ be a finite connected weighted graph, and assume $1\le \alpha \le p\le q$. In this paper, we consider the following $p$-th Yamabe type equation $$-{\Delta }_{p}u+h{u}^{q-1}=\lambda f{u}^{\alpha -1}.$$ on $G$, where ${\Delta }_p$ is the $p$-th discrete graph Laplacian, $h\le 0$ and $f>0$ are real functions defined on all vertices of $G$. Instead of the approach in [Ge3], we adopt a new approach, and prove that the above equation always has a positive solution $u>0$ for some constant $\lambda \in \mathbb{R}$. In particular, when $q=p$ our result generalizes the main theorem in [Ge3] from the case of $\alpha \ge p>1$ to the case of $1\le \alpha \le p$. It's interesting that our new approach can also work in the case of $\alpha \ge p>1$.

15 pages

References in corpus (3)

• Existence of positive solutions to some nonlinear equations on locally finite graphs
• Kazdan-Warner equation on graph in the negative case
• p-th Kazdan-Warner equation on graph

Cited by in corpus (1)

• Existence and uniqueness theorems for some semi-linear equations on locally finite graphs