Kazdan–Warner equation on graph in the negative case
By Huabin Ge. 2016
63 citations recorded by the source corpus.
Abstract Let G = ( V , E ) be a connected finite graph. In this short paper, we reinvestigate the Kazdan–Warner equation Δ u = c − h e u with c 0 on G , where h defined on V is a known function. Grigor'yan, Lin and Yang [3] showed that if the Kazdan–Warner equation has a solution, then h ‾ , the average value of h , is negative. Conversely, if h ‾ 0 , then there exists a number c − ( h ) 0 , such that the Kazdan–Warner equation is solvable for every 0 > c > c − ( h ) and it is not solvable for c c − ( h ) . Moreover, if h ≤ 0 and h ≢ 0 , then c − ( h ) = − ∞ . Inspired by Chen and Li's work [1] , we ask naturally: Is the Kazdan–Warner equation solvable for c = c − ( h ) ? In this paper, we answer the question affirmatively. We show that if c − ( h ) = − ∞ , then h ≤ 0 and h ≢ 0 . Moreover, if c − ( h ) > − ∞ , then there exists at least one solution to the Kazdan–Warner equation with c = c − ( h ) .
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/0b4fe6b680033dca2df5c15747e6ed90a924dfd3
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Citations in the corpus
- Kazdan-Warner equation on infinite graphs (2017, background): “And also there is a solution in the critical case where c = ch, which was proved by the first author of this paper Ge [10].”
- The Kazdan–Warner equation on canonically compactifiable graphs (2017, methodology): “On the one hand this concerns non-linear operators such as the p-Laplacian [22,26,31], and on the other hand non-linear equations such as the Kazdan–Warner equation [13,16].”