Calculus of Variations and Geometric Measure Theory
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A. Pinamonti - G. Stefani

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

created by stefani on 19 Jun 2021

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Submitted Paper

Inserted: 19 jun 2021
Last Updated: 19 jun 2021

Year: 2021

Abstract:

We study some semi-linear equations for the $(m,p)$-Laplacian operator on locally finite weighted graphs. We prove existence of weak solutions for all $m\in\mathbb{N}$ and $p\in(1,+\infty)$ via a variational method already known in the literature by exploiting the continuity properties of the energy functionals involved. When $m=1$, we also establish a uniqueness result in the spirit of the Brezis-Strauss Theorem. We finally provide some applications of our main results by dealing with some Yamabe-type and Kazdan-Warner-type equations on locally finite weighted graphs.

Keywords: semi-linear equations on graphs, variational method, Yamabe-type equation, Kazdan-Warner-type equation, Brezis-Strauss Theorem


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