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

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.

13 pages

References in corpus (7)

• Existence of positive solutions to some nonlinear equations on locally finite graphs
• Multiple solutions of Kazdan-Warner equation on graphs in the negative case
• Yamabe-type equations on Carnot groups
• Kazdan-Warner equation on infinite graphs
• Positive Solutions of $p$-th Yamabe Type Equations on Graphs
• Nonlinear elliptic equations on Carnot groups
• Existence and convergence of solutions for nonlinear biharmonic equations on graphs

Cited by in corpus (1)

• Nontrivial solutions for a $(p,q)$-Kirchhoff type system with concave-convex nonlinearities on locally finite graphs