Journal of the
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J. Korean Math. Soc. 2022; 59(5): 911-926

Online first article July 28, 2022      Printed September 1, 2022

https://doi.org/10.4134/JKMS.j210745

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 a 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 $h: V\rightarrow\mathbb{R}$ 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,\\
u=0,&{\rm on}&\partial\Omega,\end{array}\ri.
\end{equation*}
where $\Omega$ and $\partial\Omega$ 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*}\left\{\begin{array}{lll}
-\Delta u=f(u)+\varepsilon h, &{\rm in}& \Omega,\\
u=0, &{\rm on}&\partial\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: Primary 34B45, 35A15, 35R02

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