Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc.

Published online July 28, 2022

Copyright © The Korean Mathematical Society.

Multiple solutions of a perturbed Yamabe-type equation on graph

Yang Liu

Renmin University of China

Abstract

Let $u$ be a function on locally finite graph $G=(V, E)$ and $\Omega$ be a bounded subset of $V$. Let $\varepsilon>0$, $p>2$ and $0\leq\lambda<\lambda_1(\Omega)$ be constants, where $\lambda_1(\Omega)$ is the first eigenvalue of the discrete Laplacian, and $h: V\ra\mathbb{R}$ be a function satisfying $h\geq 0$ and $h\not\equiv 0$. We consider a perturbed Yamabe equation, say \\
\begin{equation*}\left\{\begin{array}{lll}
-\Delta u-\lambda u=|u|^{p-2}u+\varepsilon h &{\rm in}& \Omega\\[1.5ex]
u=0&{\rm on}&\p\Omega,\end{array}\ri.
\end{equation*}\\
where $\Omega$ and $\p\Omega$ denote the interior and the boundary of $\Omega$ respectively. Using variational methods,
we prove that
there exists some positive constant $\varepsilon_0>0$ such that for all $\varepsilon\in(0,\varepsilon_0)$, the above equation
has two distinct solutions. Moreover, we consider a more general nonlinear equation
\begin{equation*}\label{b3}\left\{\begin{array}{lll}
-\Delta u=f(u)+\varepsilon h &{\rm in}& \Omega\\[1.5ex]
u=0 &{\rm on}&\p\Omega,\end{array}\ri.
\end{equation*}\\
and prove similar result for certain nonlinear term $f(u)$.

Keywords: Multiple solutions, perturbed Yamabe equation, mountain-pass theorem, locally finite graph

MSC numbers: 34B45, 35A15, 35R02

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