The $1$-Yamabe equation on graph
Abstract
We study the following $1$-Yamabe equation on a connected finite graph $$\Delta_1u+g\mathrm{Sgn}(u)=h|u|^{\alpha-1}\mathrm{Sgn}(u),$$ where $\Delta_1$ is the discrete $1$-Laplacian, $\alpha>1$ and $g, h>0$ are known. We show that the above $1$-Yamabe equation always has a nontrivial solution $u\geq0$, $u\neq0$.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2017
- DOI:
- 10.48550/arXiv.1709.09867
- arXiv:
- arXiv:1709.09867
- Bibcode:
- 2017arXiv170909867G
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Analysis of PDEs
- E-Print:
- 10 pages. All comments are welcome