Yamabe type equations on graphs

By Alexander Grigorʼyan, Yong Lin, Y. Yang. 2016

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Let $G=(V,E)$ be a locally finite graph, $\Omega\subset V$ be a bounded domain, $\Delta$ be the usual graph Laplacian, and $\lambda_1(\Omega)$ be the first eigenvalue of $-\Delta$ with respect to Dirichlet boundary condition. Using the mountain pass theorem due to Ambrosetti-Rabinowitz, we prove that if $\alpha 2$, there exists a positive solution to $-\Delta u-\alpha u=|u|^{p-2}u$ in $\Omega^\circ$, $u=0$ on $\partial\Omega$, where $\Omega^\circ$ and $\partial\Omega$ denote the interior and the boundary of $\Omega$ respectively. Also we consider similar problems involving the $p$-Laplacian and poly-Laplacian by the same method. Such problems can be viewed as discrete versions of the Yamabe type equations on Euclidean space or compact Riemannian manifolds.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/73f28993850feff7e77dc4eca505a48bb5719395