# Convergence of least energy sign-changing solutions for logarithmic
Schrödinger equations on locally finite graphs

Xiaojun Chang22footnotemark: 2<EMAIL_ADDRESS>Vicen$\c{t}$iu D.
Rădulescu22footnotemark: 2<EMAIL_ADDRESS>Ru Wang22footnotemark: 2
<EMAIL_ADDRESS>Duokui Yan11footnotemark: 122footnotemark: 2
<EMAIL_ADDRESS>School of Mathematics and Statistics $\&$ Center for
Mathematics and Interdisciplinary Sciences,
Northeast Normal University, Changchun 130024, Jilin, PR China  Faculty of
Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków,
Poland  Simion Stoilow Institute of Mathematics of the Romanian Academy, 21
Calea Grivi$\c{t}$ei Street, 010702 Bucharest, Romania  School of Mathematical
Sciences, Beihang University, Beijing 100191, PR China

###### Abstract

In this paper, we study the following logarithmic Schrödinger equation

$-\Delta u+\lambda a(x)u=u\log u^{2}\ \ \ \ \mbox{ in }V$

on a connected locally finite graph $G=(V,E)$, where $\Delta$ denotes the
graph Laplacian, $\lambda>0$ is a constant, and $a(x)\geq 0$ represents the
potential. Using variational techniques in combination with the Nehari
manifold method based on directional derivative, we can prove that, there
exists a constant $\lambda_{0}>0$ such that for all $\lambda\geq\lambda_{0}$,
the above problem admits a least energy sign-changing solution $u_{\lambda}$.
Moreover, as $\lambda\to+\infty$, we prove that the solution $u_{\lambda}$
converges to a least energy sign-changing solution of the following Dirichlet
problem

$\begin{cases}-\Delta u=u\log u^{2}~{}~{}~{}&\mbox{ in }\Omega,\\\
u(x)=0~{}~{}~{}&\mbox{ on }\partial\Omega,\end{cases}$

where $\Omega=\\{x\in V:a(x)=0\\}$ is the potential well.

###### keywords:

Least energy sign-changing solutions; Logarithmic Schrödinger equations;
Locally finite graphs; Nehari manifold method.
_Mathematics Subject Classification:_ 35A15, 35R02, 35Q55, 39A12.

††footnotetext: 11footnotemark: 1Corresponding author.††footnotetext:
22footnotemark: 2These authors contributed equally to this work.

##  1 Introduction and main results

Theory of network (or graph) has a wide range of applications in various
fields such as signal processing, image processing, data clustering and
machine learning. (For example, see [1, 2, 3].) A graph $G=(V,E)$, where $V$
denotes the vertex set and $E$ denotes the edge set, is said to be locally
finite if for any $x\in V$, there are only finite $y\in V$ such that $xy\in
E$. A graph is connected if any two vertices $x$ and $y$ can be connected via
finite edges. For any $xy\in E$, we assume that its weight $\omega_{xy}>0$ and
$\omega_{xy}=\omega_{yx}$. The degree of $x\in V$ is defined by
$deg(x)=\sum_{y\sim x}\omega_{xy}$, where we write $y\sim x$ if $xy\in E$. The
distance $d(x,y)$ of two vertices $x,y\in V$ is defined by the minimal number
of edges which connect these two vertices. The measure
$\mu:V\to\mathbb{R}^{+}$ is defined to be a finite positive function on $G$.

In recent years, there have been many studies on the existence and
multiplicity of solutions to nonlinear elliptic equations on discrete graphs,
we refer to [4, 5, 6, 7, 9, 8, 10, 11] and their references. In [7],
Grigor’yan, Lin and Yang studied nonlinear Schrödinger equations

$-\Delta u+b(x)u=f(x,u)\quad\mbox{in }V$ (1.1)

on a connected locally finite graph $G$. By applying the mountain pass
theorem, they established the existence of strictly positive solutions of
(1.1) when $f$ satisfies the so-called Ambrosetti-Rabinowitz ((AR) for short)
condition, and the potential $b:V\to\mathbb{R}^{+}$ has a positive lower bound
and satisfies one of the following hypotheses:

$(B_{1})$

$b(x)\to+\infty$ as $d(x,x_{0})\to+\infty$ for some fixed $x_{0}\in V$;

$(B_{2})$

$1/b(x)\in L^{1}(V)$.

In [11], Zhang and Zhao established the existence and convergence (as
$\lambda\to+\infty$) of ground state solutions for equation (1.1), when
$b(x)=\lambda a(x)+1$ and $f(x,u)=|u|^{p-1}u$, where $a(x)\geq 0$ satisfies
$(B_{1})$ and the potential well $\Omega=\\{x\in V:a(x)=0\\}$ is a non-empty
connected and bounded domain in $V$. Similar results for $p$-Laplacian
equations and biharmonic equations on locally finite graphs can be found in
[12, 13].

In this paper, we consider the following logarithmic Schrödinger equation

$-\Delta u+\lambda a(x)u=u\log u^{2}\ \ \ \ \mbox{in }V$ (1.2)

on a connected locally finite graph $G=(V,E)$, where the parameter
$\lambda>0$. We recall that the logarithmic Schrödinger equation in Euclidean
space

$-\Delta u+\lambda b(x)u=u\log u^{2}\ \ \mbox{in }\mathbb{R}^{N}$ (1.3)

has recently received much attention, one can see [14, 15, 16, 17, 18, 19, 20,
21, 22] and references therein. Logarithmic nonlinear problems have a wide
range of applications in fields such as quantum mechanics, quantum optics,
nuclear physics, transport and diffusion phenomena, Bose-Einstein condensation
and etc. (see [23, 24, 25, 26, 27, 28]).

Different approaches have been developed to study the existence and
multiplicity of solutions for nonlinear Schrödinger equations with logarithmic
nonlinearities. Cazenave [14] worked in an Orlicz space endowed with a
Luxemburg type norm in order to make the associated energy functional of
equation (1.3) to be $C^{1}$. Squassina and Szulkin [20] studied the existence
of multiple solutions by using non-smooth critical point theory (see also [15,
16, 18]). Tanaka and Zhang [21] applied the penalization technique to study
multi-bump solutions of equation (1.3). For the idea of penalization, see also
[29, 30, 17]. In [22], Wang and Zhang proved that the ground state solution of
the power-law scalar field equations $-\Delta u+\lambda u=|u|^{p-2}u$, as
$p\downarrow 2$, converge to the ground state solution of the logarithmic-law
equation $-\Delta u=\lambda u\log u^{2}$. Recently, several results are
devoted to studying the sign-changing solutions. Chen and Tang [31]
established the existence of least energy sign-changing solutions of some
logarithmic Schrödinger equation in bounded domains of $\mathbb{R}^{N}$. Shuai
[19] obtained the existence of least energy sign-changing solutions for
equation (1.3) under different types of potentials. Zhang and Wang
investigated, in [32], the existence and concentration behaviors of sign-
changing solutions for logarithmic scalar field equations in the semiclassical
setting. Ji [33] established the existence and multiplicity of multi-bump type
nodal solutions for equation (1.3). For more studies on logarithmic nonlinear
equations, one may refer to [34, 14, 35, 15, 16, 18, 20] and their references.

The goal of this work is to show the existence of least energy sign-changing
solutions of (1.2) and their asymptotic behavior as $\lambda\to+\infty$. To
the best of our knowledge, there is no result on sign-changing solutions for
logarithmic Schrödinger problems on locally finite graphs.

In the sequel of this paper, we make the assumption that there exists a
constant $\mu_{\min}>0$ such that the measure $\mu(x)\geq\mu_{\min}>0$ for all
$x\in V$. As for the potential $a=a(x)$, we assume that:

  * $(A_{1})$

$a(x)\geq 0$ and the potential well $\Omega=\\{x\in V:a(x)=0\\}$ is a non-
empty, connected and bounded domain in V;

  * $(A_{2})$

there exists $M>0$ such that the volume of the set $D_{M}$ is finite, namely,

$Vol(D_{M})=\sum_{x\in D_{M}}\mu(x)<\infty,$

where $D_{M}=\\{x\in V:a(x)<M\\}$.

To explain our result, we first introduce some notations. For any function
$u:V\to\mathbb{R}$, the graph Laplacian of $u$ is defined by

$\Delta u(x)=\frac{1}{\mu(x)}\sum_{y\sim x}\omega_{xy}\left(u(y)-u(x)\right).$
(1.4)

The integral of $u$ over $V$ is defined by $\int_{V}ud\mu=\sum_{x\in
V}\mu(x)u(x)$, and the gradient form of the two functions $u,v$ on $V$ is
defined by

$\Gamma(u,v)(x)=\frac{1}{2\mu(x)}\sum_{y\sim
x}\omega_{xy}\left(u(y)-u(x)\right)\left(v(y)-v(x)\right).$ (1.5)

Write $\Gamma(u)=\Gamma(u,u)$, and sometimes we use $\nabla u\nabla v$ to
replace $\Gamma(u,v)$. The length of the gradient of $u$ is defined by

$|\nabla u|(x)=\sqrt{\Gamma(u)(x)}=\left(\frac{1}{2\mu(x)}\sum_{y\sim
x}\omega_{xy}\left(u(y)-u(x)\right)^{2}\right)^{1/2}.$ (1.6)

Denote by $C_{c}(V)$ the set of all functions with compact support, and let
$H^{1}(V)$ be the completion of $C_{c}(V)$ under the norm

$\|u\|_{H^{1}(V)}=\left(\int_{V}\left(|\nabla
u|^{2}+u^{2}\right)d\mu\right)^{1/2}.$

Then, $H^{1}(V)$ is a Hilbert space with the inner product

$\langle u,v\rangle=\int_{V}\left(\Gamma(u,v)+uv\right)d\mu,\quad\ \forall u,\
v\in H^{1}(V).$

We write $\|u\|_{p}=\left(\int_{V}|u|^{p}d\mu\right)^{1/p}$ for
$p\in[1,+\infty)$ and $\|u\|_{L^{\infty}}=\sup\limits_{x\in V}|u(x)|$.

For each $\lambda>0$ we introduce a space

$\mathcal{H}_{\lambda}=\left\\{u\in H^{1}(V):\int_{V}\lambda
a(x)u^{2}d\mu<+\infty\right\\}$

with norm

$\|u\|_{\mathcal{H}_{\lambda}}^{2}\dot{=}\int_{V}\left(|\nabla u|^{2}+(\lambda
a(x)+1)u^{2}\right)d\mu,$

which is induced by

$\langle u,v\rangle_{\mathcal{H}_{\lambda}}=\int_{V}\left(\Gamma(u,v)+(\lambda
a(x)+1)uv\right)d\mu,\ \forall u,\ v\in\mathcal{H}_{\lambda}.$

Clearly, $\mathcal{H}_{\lambda}$ is also a Hilbert space.

Note that equation (1.2) is formally associated with the energy functional
$J_{\lambda}:\ H^{1}(V)\to\mathbb{R}\cup\\{+\infty\\}$ given by

$J_{\lambda}(u)=\frac{1}{2}\int_{V}\left(|\nabla u|^{2}+(\lambda
a(x)+1)u^{2}\right)d\mu-\frac{1}{2}\int_{V}u^{2}\log u^{2}d\mu.$ (1.7)

Clearly, $J_{\lambda}$ fails to be $C^{1}$ in $H^{1}(V)$. In fact, for some
$G=(V,E)$ with suitable measure $\mu$, there exists $u\in H^{1}(V)$ but
$\int_{V}u^{2}\log u^{2}d\mu=-\infty$ (see [37]).

In this paper, we consider the functional $J_{\lambda}$ in (1.7) on the set

$\mathcal{D}_{\lambda}=\left\\{u\in\mathcal{H}_{\lambda}:\int_{V}u^{2}|\log
u^{2}|d\mu<\infty\right\\}.$

That is,

$J_{\lambda}(u)=\frac{1}{2}\|u\|_{\mathcal{H}_{\lambda}}^{2}-\frac{1}{2}\int_{V}u^{2}\log
u^{2}d\mu,\quad\forall u\in\mathcal{D}_{\lambda}.$

Define the Nehari manifold and sign-changing Nehari set respectively by

$\mathcal{N}_{\lambda}=\left\\{u\in\mathcal{D}_{\lambda}\setminus\\{0\\}:J^{\prime}_{\lambda}(u)\cdot
u=0\right\\},$
$\mathcal{M}_{\lambda}=\left\\{u\in\mathcal{D}_{\lambda}:u^{\pm}\neq 0\mbox{
and }J^{\prime}_{\lambda}(u)\cdot u^{+}=J^{\prime}_{\lambda}(u)\cdot
u^{-}=0\right\\},$

where $u^{+}=\max\\{u,\ 0\\}$ and $u^{-}=\min\\{u,\ 0\\}$. Clearly,
$\mathcal{N}_{\lambda}$ contains all the nontrivial solutions of equation
(1.2) and $\mathcal{M}_{\lambda}$ contains all the sign-changing solutions.
Set

$c_{\lambda}=\inf_{u\in\mathcal{N}_{\lambda}}J_{\lambda}(u),\ \ \
m_{\lambda}=\inf_{u\in\mathcal{M}_{\lambda}}J_{\lambda}(u).$

Our first result is as follows.

######  Theorem 1.1.

Suppose that $G=(V,E)$ is a connected locally finite graph and the potential
$a:V\to\mathbb{R}$ satisfies $(A_{1})$ and $(A_{2})$. Then, there exists a
constant $\lambda_{0}>0$ such that for all $\lambda\geq\lambda_{0}$, equation
(1.2) admits a least energy sign-changing solution
$u_{\lambda}\in\mathcal{D}_{\lambda}$ such that
$J_{\lambda}(u_{\lambda})=m_{\lambda}$. Moreover, $m_{\lambda}>2c_{\lambda}$.

We recall that $D\subset V$ is a bounded domain if the distance $d(x,y)$
between any $x,y\in D$ is uniformly bounded. The boundary of $D$ is defined by

$\partial D\dot{=}\\{y\not\in D:\mbox{there exists}~{}x\in D\mbox{ such that
}xy\in E\\}$

and the interior of $D$ is denoted by $D^{\circ}$. Obviously, $D^{\circ}=D$.
Set $\Omega=\\{x\in V:a(x)=0\\}$. Let $H_{0}^{1}(\Omega)$ be the completion of
$C_{c}(\Omega)$ under the norm

$\|u\|_{H_{0}^{1}(\Omega)}=\left(\int_{\Omega\cup\partial\Omega}|\nabla
u|^{2}d\mu+\int_{\Omega}u^{2}d\mu\right)^{1/2}.$

Then, $H_{0}^{1}(\Omega)$ is a Hilbert space.

We consider the following Dirichlet problem

$\begin{cases}-\Delta u=u\log u^{2}~{}~{}~{}&\mbox{ in }\Omega,\\\
u(x)=0~{}~{}~{}&\mbox{ on }\partial\Omega.\end{cases}$ (1.8)

The energy functional $J_{\Omega}:H_{0}^{1}(\Omega)\to\mathbb{R}$ associated
with $(\ref{eq-eq2})$ is given by

$J_{\Omega}(u)\dot{=}\frac{1}{2}\|u\|_{H_{0}^{1}(\Omega)}^{2}-\frac{1}{2}\int_{\Omega}u^{2}\log
u^{2}d\mu,\quad\forall u\in H_{0}^{1}(\Omega).$

Define

$\mathcal{N}_{\Omega}=\left\\{u\in
H_{0}^{1}(\Omega)\setminus\\{0\\}:J^{\prime}_{\Omega}(u)\cdot u=0\right\\},$
$\mathcal{M}_{\Omega}=\left\\{u\in H_{0}^{1}(\Omega):u^{\pm}\neq 0\mbox{ and
}J^{\prime}_{\Omega}(u)\cdot u^{+}=J^{\prime}_{\Omega}(u)\cdot
u^{-}=0\right\\}.$

Set

$c_{\Omega}=\inf_{u\in\mathcal{N}_{\Omega}}J_{\lambda}(u),\ \ \
m_{\Omega}=\inf_{u\in\mathcal{M}_{\Omega}}J_{\Omega}(u).$

Similar to Theorem 1.1, problem $(\ref{eq-eq2})$ also has a least energy sign-
changing solution.

######  Theorem 1.2.

Let $G=(V,E)$ be a connected locally finite graph. Assume $\Omega=\\{x\in
V:a(x)=0\\}$ is a non-empty, connected and bounded domain in V. Then problem
(1.8) admits a least energy sign-changing solution $u_{0}\in
H_{0}^{1}(\Omega)$ such that $J_{\Omega}(u_{\Omega})=m_{\Omega}$. Moreover,
$m_{\Omega}>2c_{\Omega}$.

Finally, we show the convergence of $u_{\lambda}$ as $\lambda\to+\infty$.

######  Theorem 1.3.

Under the assumptions of Theorem 1.1, we conclude that for any sequence
$\lambda_{k}\to+\infty$, up to a subsequence, the corresponding least energy
sign-changing solution $u_{\lambda_{k}}$ of equation $(\ref{eq-eq1})$
converges in $H^{1}(V)$ to a least energy sign-changing solution of problem
$(\ref{eq-eq2})$.

One of the main challenges in proving Theorem 1.1-1.3 is to deal with the
logarithmic term in equation (1.2). In Euclidean space, logarithmic Sobolev
inequality is important in studying the logarithmic Schrödinger equation (see
[30, 19, 20] etc.). While, on discrete graphs, the logarithmic Sobolev
inequality is only available under a positive curvature condition, which
requires the measure $\mu$ to be finite (see [36] for details). In our case,
the measure $\mu$ has a uniform positive lower bound, which violates the
positive curvature condition. To overcome this difficulty, we will develop new
and delicate arguments not relying on the logarithmic Sobolev inequality.

In addition, in view of the associated energy functional with (1.2) is not
well-defined, inspired by ideas in [19, 22], we will restrict $u^{2}\log
u^{2}\in L^{1}(V)$. New challenge arises since the techniques in [19, 22] are
not applicable here because the graph Laplacian operator is non-local. To be
precise, in [19], denoting by $I$ the corresponding energy functional, we know
that the following decomposition

$\displaystyle
I(u)=I(u^{+})+I(u^{-}),\quad\big{<}I^{\prime}(u),u\big{>}=\big{<}I^{\prime}(u^{+}),u^{+}\big{>}+\big{<}I^{\prime}(u^{-}),u^{-}\big{>},$
(1.9)

plays a key role in studying nodal solutions. However, in our case, such a
decomposition does not hold. Actually, by a direct computation, it follows
that for each $u\in\mathcal{D}_{\lambda}\setminus\\{0\\}$,

$\displaystyle
J_{\lambda}(u)=J_{\lambda}(u^{+})+J_{\lambda}(u^{-})-\frac{1}{2}K_{V}(u),$
$\displaystyle J^{\prime}_{\lambda}(u)\cdot
u^{\pm}=J^{\prime}_{\lambda}(u^{\pm})\cdot u^{\pm}-\frac{1}{2}K_{V}(u),$

where $K_{V}(u)=\sum\limits_{x\in V}\sum\limits_{y\sim
x}\omega_{xy}\left[u^{+}(x)u^{-}(y)+u^{-}(x)u^{+}(y)\right]<0$, see Section 2
for details. Clearly, $J(u)\neq J(u^{+})+J(u^{-})$ and
$\big{<}J^{\prime}(u),u\big{>}\neq\big{<}J^{\prime}(u^{+}),u^{+}\big{>}+\big{<}J^{\prime}(u^{-}),u^{-}\big{>}$,
which imply that (1.9) fails. Motivated by [38, 39], we will develop new
variational arguments involving nonlocal operator based on directional
derivative to study (1.2).

The paper is organized as follows. In Section 2, we introduce some notations,
definitions and preliminary lemmas. In Section 3, we apply the Nehari manifold
method to prove the existence of least energy sign-changing solution of
equation $(\ref{eq-eq1})$ and the Dirichlet problem $(\ref{eq-eq2})$. In
Section 4, we give the proof of Theorem 1.3.

##  2 Some preliminary results

###  2.1 Some definitions

To prove Theorem 1.1, we introduce the following definitions.

######  Definition 2.1.

Given $u\in\mathcal{D}_{\lambda}$ and $\phi\in C_{c}(V)$, the derivative of
$J_{\lambda}$ in the direction $\phi$ at $u$, denoted by
$J^{\prime}_{\lambda}(u)\cdot\phi$, is defined as $\lim_{t\to
0^{+}}\frac{1}{t}\left[J_{\lambda}(u+t\phi)-J_{\lambda}(u)\right]$.

######  Definition 2.2.

  * (1)

For $u,v\in\mathcal{D}_{\lambda}$, we define

$J_{\lambda}^{\prime}(u)\cdot v:=\int_{V}\left(\Gamma(u,v)+\lambda
a(x)uv\right)d\mu-\int_{V}uv\log u^{2}d\mu.$

Clearly, $\int_{V}uv\log u^{2}d\mu$ is well-defined for
$u,v\in\mathcal{D}_{\lambda}$.

  * (2)

We say that $u\in\mathcal{H}_{\lambda}$ is a critical point of $J_{\lambda}$
if $u\in\mathcal{D}_{\lambda}$ and $J_{\lambda}^{\prime}(u)\cdot v=0$ for all
$v\in\mathcal{D}_{\lambda}$. We also say that $d_{\lambda}\in\mathbb{R}$ is a
critical value for $J_{\lambda}$ if there exists a critical point
$u\in\mathcal{H}_{\lambda}$ such that $J_{\lambda}(u)=d_{\lambda}$.

It is easily seen that, $u$ is a weak solution to equation (1.2) if and only
if $u$ is a critical point of $J_{\lambda}$.

Note that, for any $0<\varepsilon<1$, there exists $C_{\varepsilon}>0$ such
that

$|u^{2}\log u^{2}|\leq
C_{\varepsilon}(|u|^{2-\varepsilon}+|u|^{2+\varepsilon}).$

Since $H^{1}(\Omega)\hookrightarrow L^{p}(\Omega)$ is compact for
$p\in[1,+\infty]$, we have $J_{\Omega}\in C^{1}(H_{0}^{1}(\Omega),\mathbb{R})$
and

$J^{\prime}_{\Omega}(u)\cdot v=\int_{\Omega\cup\partial\Omega}\nabla u\nabla
vd\mu-\int_{\Omega}uv\log u^{2}d\mu,\forall u,v\in H_{0}^{1}(\Omega).$

Clearly, $u$ is a weak solution to problem (1.8) if and only if $u$ is a
critical point of $J_{\Omega}$.

######  Lemma 2.3.

If $u\in\mathcal{D}_{\lambda}$ is a weak solution of (1.2), then $u$ is a
point-wise solution of (1.2).

###### Proof.

If $u\in\mathcal{D}_{\lambda}$ is a weak solution of (1.2), then for any
$\varphi\in\mathcal{D}_{\lambda}$, there holds

$\int_{V}\left(\Gamma(u,\varphi)+\lambda
a(x)u\varphi\right)d\mu=\int_{V}u\varphi\log u^{2}d\mu.$

Using $C_{c}(V)$ is dense in $\mathcal{D}_{\lambda}$ and $\omega_{x,y}$ is
symmetric, for any $\varphi\in C_{c}(V)$, by integration by parts, we have

$\displaystyle\int_{V}\Gamma(u,\varphi)d\mu=$
$\displaystyle\frac{1}{2}\sum_{x\in V}\sum_{y\sim
x}\omega_{xy}\left(u(y)-u(x)\right)\left(\varphi(y)-\varphi(x)\right)$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\sum_{y\in V}\sum_{x\sim
y}\omega_{xy}\left(u(y)-u(x)\right)\varphi(x)-\frac{1}{2}\sum_{x\in
V}\sum_{y\sim x}\omega_{xy}\left(u(y)-u(x)\right)\varphi(x)$ $\displaystyle=$
$\displaystyle-\sum_{x\in V}\sum_{y\sim
x}\omega_{xy}\left(u(y)-u(x)\right)\varphi(x)$ $\displaystyle=$
$\displaystyle-\int_{V}\Delta u\varphi d\mu,$

which gives

$\int_{V}\left(-\Delta u+\lambda a(x)u\right)\varphi d\mu=\int_{V}u\varphi\log
u^{2}d\mu,\ \ \forall\varphi\in C_{c}(V).$ (2.1)

For any fixed $y\in V$, take a test function $\varphi:V\to\mathbb{R}$ in (2.1)
with

$\varphi(x)=\begin{cases}1,~{}&x=y,\\\ 0,~{}&x\neq y.\end{cases}$

Clearly, $\varphi\in\mathcal{D}_{\lambda}$ and $-\Delta u(y)+\lambda
a(y)u(y)-u(y)\log\left(u(y)\right)^{2}=0$. Since $y$ is arbitrary, we conclude
that $u$ is a point-wise solution of (1.2). ∎

Similarly, we obtain

######  Lemma 2.4.

If $u\in H_{0}^{1}(\Omega)$ is a weak solution of (1.8), then $u$ is a point-
wise solution of (1.8).

Next, we have the following observations:

$\displaystyle\int_{V}\Gamma(u^{+}+u^{-})d\mu=\int_{V}\Gamma(u^{+})d\mu+\int_{V}\Gamma(u^{-})d\mu-
K_{V}(u),$ (2.2)
$\displaystyle\int_{V}\Gamma(u^{+}+u^{-},u^{+})d\mu=\int_{V}\Gamma(u^{+})d\mu-\frac{1}{2}K_{V}(u),$
(2.3)
$\displaystyle\int_{V}\Gamma(u^{+}+u^{-},u^{-})d\mu=\int_{V}\Gamma(u^{-})d\mu-\frac{1}{2}K_{V}(u).$
(2.4)

Then, for each $u\in\mathcal{D}_{\lambda}$, we have

$\displaystyle
J_{\lambda}(u)=J_{\lambda}(u^{+})+J_{\lambda}(u^{-})-\frac{1}{2}K_{V}(u),$
$\displaystyle J^{\prime}_{\lambda}(u)\cdot
u^{\pm}=J^{\prime}_{\lambda}(u^{\pm})\cdot u^{\pm}-\frac{1}{2}K_{V}(u),$

and for each $u\in H_{0}^{1}(\Omega)$,

$\displaystyle
J_{\Omega}(u)=J_{\Omega}(u^{+})+J_{\Omega}(u^{-})-\frac{1}{2}K_{\Omega}(u),$
$\displaystyle J^{\prime}_{\Omega}(u)\cdot
u^{\pm}=J^{\prime}_{\Omega}(u^{\pm})\cdot u^{\pm}-\frac{1}{2}K_{\Omega}(u),$

where
$K_{\Omega}(u):=\sum\limits_{x\in\Omega\cup\partial\Omega}\sum\limits_{y\sim
x}\omega_{xy}\left[u^{+}(x)u^{-}(y)+u^{-}(x)u^{+}(y)\right]$.

###  2.2 Sobolev embedding

In this subsection, we establish a Sobolev embedding result.

######  Lemma 2.5.

If $\mu(x)\geq\mu_{\min}>0$ and $a(x)$ satisfies $(A_{1})-(A_{2})$, then there
exist a constant $\lambda_{0}>0$ such that, for all $\lambda\geq\lambda_{0}$,
the space $\mathcal{H}_{\lambda}$ is compactly embedded into $L^{p}(V)$ for
all $2\leq p\leq+\infty$.

###### Proof.

For all $\lambda>0$, at any vertex $x_{0}\in V$, by $(A_{1})$ we have

$\displaystyle\|u\|_{\mathcal{H}_{\lambda}}^{2}=$
$\displaystyle\int_{V}\left(|\nabla u|^{2}+(\lambda
a(x)+1)u^{2}\right)d\mu\geq\int_{V}u^{2}d\mu\geq\mu_{\min}u^{2}(x_{0}),$

which implies that
$|u(x_{0})|\leq\sqrt{\frac{1}{\mu_{\min}}}\|u\|_{\mathcal{H}_{\lambda}}$. Thus
$\mathcal{H}_{\lambda}\hookrightarrow L^{\infty}(V)$ continuously. Hence,
using interpolation gives that $\mathcal{H}_{\lambda}\hookrightarrow L^{p}(V)$
continuously for all $2\leq p\leq\infty$. Assuming $\\{u_{k}\\}$ is bounded in
$\mathcal{H}_{\lambda}$, we have that, up to a subsequence,
$u_{k}\rightharpoonup u$ in $\mathcal{H}_{\lambda}$. In particular,
$\\{u_{k}\\}\subset\mathcal{H}_{\lambda}$ is also bounded in $L^{2}(V)$ and by
the weak convergence in $L^{2}(V)$ it follows that, for any $\varphi\in
L^{2}(V)$,

$\lim_{k\to\infty}\int_{V}(u_{k}-u)\varphi d\mu=\lim_{k\to\infty}\sum_{x\in
V}\mu(x)\left(u_{k}(x)-u(x)\right)\varphi(x)=0.$ (2.5)

Take any $x_{0}\in V$ and let

$\varphi_{0}(x)=\begin{cases}1,~{}x=x_{0},\\\ 0,~{}x\neq x_{0}.\end{cases}$

Obviously, $\varphi_{0}(x)\in L^{2}(V)$. By substituting $\varphi_{0}$ into
(2.5), we get $\lim_{k\to\infty}u_{k}(x)=u(x)$ for any fixed $x\in V$.

Since $u_{k}$ is bounded in $\mathcal{H}_{\lambda}$ and
$u\in\mathcal{H}_{\lambda}$, there exists $C_{1}>0$ such that

$\lambda\int_{V}a(x)(u_{k}-u)^{2}d\mu\leq C_{1}.$

We claim that, up to a subsequence,

$\lim_{k\to+\infty}\int_{V}(u_{k}-u)^{2}d\mu=0.$

In fact, since $a(x)$ satisfies $(A_{2})$, there exists some $M>0$ such that

$\displaystyle\int_{V}(u_{k}-u)^{2}d\mu=$
$\displaystyle\int_{D_{M}}(u_{k}-u)^{2}d\mu+\int_{V\setminus
D_{M}}(u_{k}-u)^{2}d\mu$ $\displaystyle\leq$
$\displaystyle\int_{D_{M}}(u_{k}-u)^{2}d\mu+\int_{V\setminus
D_{M}}\frac{1}{\lambda M}\lambda a(x)(u_{k}-u)^{2}d\mu$ $\displaystyle\leq$
$\displaystyle\int_{D_{M}}(u_{k}-u)^{2}d\mu+\frac{C_{1}}{\lambda M}.$

Then, for all $\varepsilon>0$, there exists $\lambda_{0}>0$ such that when
$\lambda>\lambda_{0}$, we have $\frac{C_{1}}{\lambda M}<\varepsilon$.
Moreover, up to a subsequence, we have

$\lim_{k\to+\infty}\int_{D_{M}}(u_{k}-u)^{2}d\mu=0.$

Hence the claim holds. In view of
$\|u_{k}-u\|_{\infty}^{2}\leq\frac{1}{\mu_{\min}}\int_{V}|u_{k}-u|^{2}d\mu$,
for any $2<p<\infty$, we deduce

$\int_{V}|u_{k}-u|^{p}d\mu\leq\left(\frac{1}{\mu_{\min}}\right)^{\frac{p-2}{2}}\left(\int_{V}|u_{k}-u|^{2}d\mu\right)^{\frac{p}{2}}.$

Therefore, up to a subsequence, $u_{k}\to u$ in $L^{p}(V)$ for all $2\leq
p\leq+\infty$. ∎

##  3 Existence of least energy sign-changing solutions

This section is devoted to proving that equation (1.2), as well as $(\ref{eq-
eq2})$, admits a least energy sign-changing solution.

The following result will be useful.

######  Lemma 3.1.

For all $u\in\mathcal{M}_{\lambda}$ and $s,t>0$, there holds

$J_{\lambda}(u)\geq J_{\lambda}(su^{+}+tu^{-}).$

The ”=” holds if and only if $s=t=1$.

###### Proof.

For any $u\in\mathcal{M}_{\lambda}$,

$\displaystyle J_{\lambda}(u)=$ $\displaystyle
J_{\lambda}(u)-\frac{1}{2}J^{\prime}_{\lambda}(u)\cdot
u^{+}-\frac{1}{2}J^{\prime}_{\lambda}(u)\cdot u^{-}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\|u^{+}\|_{\mathcal{H}_{\lambda}}^{2}-\frac{1}{2}\int_{V}|u^{+}|^{2}\log|u^{+}|^{2}d\mu\right)-\left(\frac{1}{2}\|u^{+}\|_{\mathcal{H}_{\lambda}}^{2}-\frac{1}{2}\int_{V}|u^{+}|^{2}\log|u^{+}|^{2}d\mu-\frac{1}{2}\|u^{+}\|_{2}^{2}\right)$
$\displaystyle+\left(\frac{1}{2}\|u^{-}\|_{\mathcal{H}_{\lambda}}^{2}-\frac{1}{2}\int_{V}|u^{-}|^{2}\log|u^{-}|^{2}d\mu\right)-\left(\frac{1}{2}\|u^{-}\|_{\mathcal{H}_{\lambda}}^{2}-\frac{1}{2}\int_{V}|u^{-}|^{2}\log|u^{-}|^{2}d\mu-\frac{1}{2}\|u^{+}\|_{2}^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\|u^{+}\|_{2}^{2}+\frac{1}{2}\|u^{-}\|_{2}^{2}.$

For $s,t>0$, by (2.2) we obtain

$\displaystyle\int_{V}\Gamma(su^{+}+tu^{-})d\mu=\int_{V}\Gamma(su^{+})d\mu+\int_{V}\Gamma(tu^{-})-stK_{V}(u).$

Hence,

$\displaystyle J_{\lambda}(su^{+}+tu^{-})$ $\displaystyle=$ $\displaystyle
J_{\lambda}(su^{+})+J_{\lambda}(tu^{-})-\frac{st}{2}K_{V}(u)$ $\displaystyle=$
$\displaystyle s^{2}J_{\lambda}(u^{+})-\frac{1}{2}s^{2}\log
s^{2}\|u^{+}\|_{2}^{2}+t^{2}J_{\lambda}(u^{-})-\frac{1}{2}t^{2}\log
t^{2}\|u^{-}\|_{2}^{2}-\frac{st}{2}K_{V}(u)$ $\displaystyle=$ $\displaystyle
s^{2}\left[J_{\lambda}(u^{+})-\frac{1}{2}J^{\prime}_{\lambda}(u)\cdot
u^{+}\right]-\frac{1}{2}s^{2}\log
s^{2}\|u^{+}\|_{2}^{2}+t^{2}\left[J_{\lambda}(u^{-})-\frac{1}{2}J^{\prime}_{\lambda}(u)\cdot
u^{-}\right]$ $\displaystyle-\frac{1}{2}t^{2}\log
t^{2}\|u^{-}\|_{2}^{2}-\frac{st}{2}K_{V}(u)$ $\displaystyle=$ $\displaystyle
s^{2}\left[J_{\lambda}(u^{+})-\frac{1}{2}J^{\prime}_{\lambda}(u^{+})\cdot
u^{+}+\frac{1}{4}K_{V}(u)\right]-\frac{1}{2}s^{2}\log s^{2}\|u^{+}\|_{2}^{2}$
$\displaystyle+t^{2}\left[J_{\lambda}(u^{-})-\frac{1}{2}J^{\prime}_{\lambda}(u^{-})\cdot
u^{-}+\frac{1}{4}K_{V}(u)\right]-\frac{1}{2}t^{2}\log
t^{2}\|u^{-}\|_{2}^{2}-\frac{st}{2}K_{V}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(s^{2}-s^{2}\log
s^{2})\|u^{+}\|_{2}^{2}+\frac{1}{2}(t^{2}-t^{2}\log
t^{2})\|u^{-}\|_{2}^{2}+\frac{(s-t)^{2}}{4}K_{V}(u).$

Therefore, defining $f(\tau)=\tau^{2}-\tau^{2}\log\tau^{2}-1$ for any
$\tau\geq 0$, we have

$\displaystyle J_{\lambda}(su^{+}+tu^{-})-J_{\lambda}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(s^{2}-s^{2}\log
s^{2}-1)\|u^{+}\|_{2}^{2}+\frac{1}{2}(t^{2}-t^{2}\log
t^{2}-1)\|u^{-}\|_{2}^{2}+\frac{(s-t)^{2}}{4}K_{V}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}f(s)\|u^{+}\|_{2}^{2}+\frac{1}{2}f(t)\|u^{-}\|_{2}^{2}+\frac{(s-t)^{2}}{4}K_{V}(u).$

Since $f(0)=-1$, $f(1)=0$ and $f(\tau)<0$ if $\tau\neq 1$,
$\frac{(s-t)^{2}}{4}K_{V}(u)<0$ for any $s\neq t$, the conclusions follow.∎

Next we show $\mathcal{M}_{\lambda}\neq\emptyset$.

######  Lemma 3.2.

If $u\in\mathcal{D}_{\lambda}\setminus\\{0\\}$ with $u^{\pm}\neq 0$, then
there exists a unique positive number pair $(s_{u},t_{u})$ satisfying
$s_{u}u^{+}+t_{u}u^{-}\in\mathcal{M}_{\lambda}$.

###### Proof.

For $s,t>0$, by (2.3) and (2.4), we get

$\displaystyle\int_{V}\Gamma(su^{+}+tu^{-},su^{+})d\mu=\int_{V}\Gamma(su^{+})d\mu-\frac{st}{2}K_{V}(u)$

and

$\displaystyle\int_{V}\Gamma(su^{+}+tu^{-},tu^{-})d\mu=\int_{V}\Gamma(tu^{-})d\mu-\frac{st}{2}K_{V}(u).$

Let

$\displaystyle g_{1}(s,t)\doteq$ $\displaystyle
J^{\prime}_{\lambda}(su^{+}+tu^{-})\cdot(su^{+})$ $\displaystyle=$
$\displaystyle
s^{2}\|u^{+}\|_{\mathcal{H}_{\lambda}}^{2}-s^{2}\int_{V}|u^{+}|^{2}\log|u^{+}|^{2}d\mu-s^{2}\log
s^{2}\|u^{+}\|_{2}^{2}-s^{2}\|u^{+}\|_{2}^{2}-\frac{st}{2}K_{V}(u)$

and

$\displaystyle g_{2}(s,t)\doteq$ $\displaystyle
J^{\prime}_{\lambda}(su^{+}+tu^{-})\cdot(tu^{-})$ $\displaystyle=$
$\displaystyle
t^{2}\|u^{-}\|_{\mathcal{H}_{\lambda}}^{2}-t^{2}\int_{V}|u^{-}|^{2}\log|u^{-}|^{2}d\mu-t^{2}\log
t^{2}\|u^{-}\|_{2}^{2}-t^{2}\|u^{-}\|_{2}^{2}-\frac{st}{2}K_{V}(u).$

We can see that there exists $r_{1}>0$ small enough and $R_{1}>0$ large enough
such that

$\displaystyle g_{1}(s,s)>0,\ g_{2}(s,s)>0\ \mbox{for all }s\in(0,r_{1}),$
$\displaystyle g_{1}(s,s)<0,\ g_{2}(s,s)<0\ \mbox{for all
}s\in(R_{1},+\infty).$

Hence, there exist $0<r<R$ such that

$\displaystyle g_{1}(r,t)>0,\ g_{1}(R,t)<0\ \mbox{for all }t\in[r,R],$
$\displaystyle g_{2}(s,r)>0,\ g_{2}(s,R)<0\ \mbox{for all }s\in[r,R].$

Applying Miranda’s theorem [40], there exist some $s_{u},t_{u}\in[r,R]$ such
that $g_{1}(s_{u},t_{u})=g_{2}(s_{u},t_{u})=0$, which implies that
$s_{u}u^{+}+t_{u}u^{-}\in\mathcal{M}_{\lambda}$.

In what follows, we prove the uniqueness of $(s_{u},t_{u})$. If
$u\in\mathcal{M}_{\lambda}$, then

$0=J^{\prime}_{\lambda}(u)\cdot u^{+}=J^{\prime}_{\lambda}(u^{+})\cdot
u^{+}-\frac{1}{2}K_{V}(u)$ (3.1)

and

$0=J^{\prime}_{\lambda}(u)\cdot u^{-}=J^{\prime}_{\lambda}(u^{-})\cdot
u^{-}-\frac{1}{2}K_{V}(u).$ (3.2)

We claim that $(s_{u},t_{u})=(1,1)$ is the unique pair such that
$s_{u}u^{+}+t_{u}u^{-}\in\mathcal{M}_{\lambda}$. Indeed, if
$(s_{u},t_{u})=(1,1)$ satisfies
$s_{u}u^{+}+t_{u}u^{-}\in\mathcal{M}_{\lambda}$, without loss of generality,
we assume that $0<s_{u}\leq t_{u}$. Then

$\displaystyle 0=$ $\displaystyle
J^{\prime}_{\lambda}(s_{u}u^{+}+t_{u}u^{-})\cdot(s_{u}u^{+})$ (3.3)
$\displaystyle=$ $\displaystyle s_{u}^{2}J^{\prime}_{\lambda}(u^{+})\cdot
u^{+}-s_{u}^{2}\log s_{u}^{2}\|u^{+}\|_{2}^{2}-\frac{s_{u}t_{u}}{2}K_{V}(u)$
$\displaystyle\geq$ $\displaystyle s_{u}^{2}J^{\prime}_{\lambda}(u^{+})\cdot
u^{+}-s_{u}^{2}\log s_{u}^{2}\|u^{+}\|_{2}^{2}-\frac{s_{u}^{2}}{2}K_{V}(u)$

and

$\displaystyle 0=$ $\displaystyle
J^{\prime}_{\lambda}(s_{u}u^{+}+t_{u}u^{-})\cdot(t_{u}u^{-})$ (3.4)
$\displaystyle=$ $\displaystyle t_{u}^{2}J^{\prime}_{\lambda}(u^{-})\cdot
u^{-}-t_{u}^{2}\log t_{u}^{2}\|u^{-}\|_{2}^{2}-\frac{s_{u}t_{u}}{2}K_{V}(u)$
$\displaystyle\leq$ $\displaystyle t_{u}^{2}J^{\prime}_{\lambda}(u^{-})\cdot
u^{-}-t_{u}^{2}\log t_{u}^{2}\|u^{-}\|_{2}^{2}-\frac{t_{u}^{2}}{2}K_{V}(u).$

Together with (3.1) and (3.3), we get

$s_{u}^{2}\log s_{u}^{2}\int_{V}|u^{+}|^{2}d\mu\geq 0,$

Similarly, by (3.2) and (3.4), we can deduce that

$t_{u}^{2}\log t_{u}^{2}\int_{V}|u^{-}|^{2}d\mu\leq 0,$

which implies that $s_{u}\geq 1$ and $t_{u}\leq 1$. In view of $0<s_{u}\leq
t_{u}$, it follows that $s_{u}=t_{u}=1$.

If $u\not\in\mathcal{M}_{\lambda}$, let $(s_{1},t_{1})$ and $(s_{2},t_{2})$ be
two different positive pairs such that
$v_{i}:=s_{i}u^{+}+t_{i}u^{-}\in\mathcal{M}_{\lambda},i=1,2$, which shows that

$\frac{s_{2}}{s_{1}}v_{1}^{+}+\frac{t_{2}}{t_{1}}v_{1}^{-}=v_{2}\in\mathcal{M}_{\lambda}.$

By similar analysis as above, we obtain
$\frac{s_{2}}{s_{1}}=\frac{s_{2}}{s_{1}}=1$. This implies that
$(s_{1},t_{1})=(s_{2},t_{2})$. ∎

######  Lemma 3.3.

Let $u\in\mathcal{D}_{\lambda}$ with $u^{\pm}\neq 0$ such that
$J^{\prime}_{\lambda}(u)\cdot u^{\pm}\leq 0$. Then the unique pair
$(s_{u},t_{u})$ obtained in Lemma 3.2 satisfies $s_{u},t_{u}\in(0,1]$. In
particular, the ”=” holds if and only if $s_{u}=t_{u}=1$.

###### Proof.

Without loss of generality, we assume that $0<t_{u}\leq s_{u}$. Since
$s_{u}u^{+}+t_{u}u^{-}\in\mathcal{M}_{\lambda}$, we have

$\displaystyle
0=J^{\prime}_{\lambda}(s_{u}u^{+}+t_{u}u^{-})\cdot(s_{u}u^{+})=s_{u}^{2}J^{\prime}_{\lambda}(u^{+})\cdot
u^{+}-s_{u}^{2}\log s_{u}^{2}\|u^{+}\|_{2}^{2}-\frac{s_{u}t_{u}}{2}K_{V}(u).$
(3.5)

Note that $K_{V}(x,y)<0$, using $J^{\prime}_{\lambda}(u)\cdot u^{+}\leq 0$ and
(3.5), we can deduce that

$\displaystyle 0\leq$ $\displaystyle
s_{u}^{2}\left(J^{\prime}_{\lambda}(u^{+})\cdot
u^{+}-\frac{1}{2}K_{V}(x,y)\right)-s_{u}^{2}\log s_{u}^{2}\|u^{+}\|_{2}^{2}$
$\displaystyle=$ $\displaystyle s_{u}^{2}J^{\prime}_{\lambda}(u)\cdot
u^{+}-s_{u}^{2}\log s_{u}^{2}\|u^{+}\|_{2}^{2}$ $\displaystyle\leq$
$\displaystyle-s_{u}^{2}\log s_{u}^{2}\|u^{+}\|_{2}^{2},$

which implies that $0<s_{u}\leq 1$. Therefore, $0<t_{u}\leq s_{u}\leq 1$. ∎

Similarly, we have

######  Lemma 3.4.

If $u\in H_{0}^{1}({\Omega})\setminus\\{0\\}$ with $u^{\pm}\neq 0$, then there
exists a unique positive number pair $(s_{u},t_{u})$ satisfying
$s_{u}u^{+}+t_{u}u^{-}\in\mathcal{M}_{\Omega}$.

######  Lemma 3.5.

Let $u\in H_{0}^{1}(\Omega)$ with $u^{\pm}\neq 0$ such that
$J^{\prime}_{\Omega}(u)\cdot u^{\pm}\leq 0$. Then the unique pair
$(s_{u},t_{u})$ obtained in Lemma 3.4 satisfies $s_{u},t_{u}\in(0,1]$. In
particular, the ”=” holds if and only if $s_{u}=t_{u}=1$.

Now we prove that the minimizer of $J_{\lambda}$ on $\mathcal{M}_{\lambda}$ is
achieved.

######  Lemma 3.6.

Supposed $(A_{1})$ and $(A_{2})$ hold. Then $m_{\lambda}>0$ is achieved.

###### Proof.

Taking a minimizing sequence $\\{u_{k}\\}\subset\mathcal{M}_{\lambda}$ of
$J_{\lambda}$ yields

$\displaystyle\lim_{k\to+\infty}J_{\lambda}(u_{k})=$
$\displaystyle\lim_{k\to+\infty}\left[J_{\lambda}(u_{k})-\frac{1}{2}J^{\prime}_{\lambda}(u_{k})\cdot
u_{k}^{+}-\frac{1}{2}J^{\prime}_{\lambda}(u_{k})\cdot u_{k}^{-}\right]$ (3.6)
$\displaystyle=$
$\displaystyle\lim_{k\to+\infty}\left[J_{\lambda}(u_{k}^{+})-\frac{1}{2}J^{\prime}_{\lambda}(u_{k}^{+})\cdot
u_{k}^{+}+J_{\lambda}(u_{k}^{-})-J^{\prime}_{\lambda}(u_{k}^{-})\cdot
u_{k}^{-}\right]$ $\displaystyle=$
$\displaystyle\lim_{k\to+\infty}\left(\frac{1}{2}\|u_{k}^{+}\|_{2}^{2}+\frac{1}{2}\|u_{k}^{-}\|_{2}^{2}\right)=m_{\lambda}.$

By Lemma 2.5, the Hölder’s inequality and Young inequality, for any
$\varepsilon\in(0,1)$, there exist
$C_{\varepsilon},C_{\varepsilon}^{\prime},C_{\varepsilon}^{\prime\prime}>0$
such that

$\displaystyle\int_{V}|u_{k}^{\pm}|^{2}\log|u_{k}^{\pm}|^{2}d\mu\leq$
$\displaystyle\int_{V}(|u_{k}^{\pm}|^{2}\log|u_{k}^{\pm}|^{2})^{+}d\mu\leq
C_{\varepsilon}\int_{V}|u_{k}^{\pm}|^{2+\varepsilon}d\mu$ $\displaystyle\leq$
$\displaystyle
C_{\varepsilon}\left(\int_{V}|u_{k}^{\pm}|^{2}d\mu\right)^{\frac{1}{2}}\left(\int_{V}|u_{k}^{\pm}|^{2(1+\varepsilon)}d\mu\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C_{\varepsilon}^{\prime}\|u_{k}^{\pm}\|_{2}\|u_{k}^{\pm}\|_{\mathcal{H}_{\lambda}}^{1+\varepsilon}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\|u_{k}^{\pm}\|_{\mathcal{H}_{\lambda}}^{2}+C_{\varepsilon}^{\prime\prime}\|u_{k}^{\pm}\|_{2}^{\frac{2}{1-\varepsilon}}.$

Since $\\{u_{k}\\}\subset\mathcal{M}_{\lambda}$, we deduce that

$\displaystyle\|u_{k}^{\pm}\|_{\mathcal{H}_{\lambda}}^{2}-\frac{1}{2}K_{V}^{k}(x,y)\leq\frac{1}{2}\|u_{k}^{\pm}\|_{\mathcal{H}_{\lambda}}^{2}+C_{\varepsilon}^{\prime\prime}\|u_{k}^{\pm}\|_{2}^{\frac{2}{1-\varepsilon}}+\|u_{k}^{\pm}\|_{2}^{2},$
(3.7)

where $K_{V}^{k}(u)=\sum\limits_{x\in V}\sum\limits_{y\sim
x}\left[u_{k}^{+}(x)u_{k}^{-}(y)+u_{k}^{-}(x)u_{k}^{+}(y)\right]$. This
together with (3.6) implies that $\\{u_{k}^{\pm}\\}$ is bounded in
$\mathcal{H}_{\lambda}$ and $\\{u_{k}\\}$ is also bounded in
$\mathcal{H}_{\lambda}$. Then, there exists $\lambda_{0}>0$ such that
$\lambda\geq\lambda_{0}$, by Lemma 2.5, there exists
$u_{\lambda}\in\mathcal{H}_{\lambda}$ such that

$\begin{cases}u_{k}\rightharpoonup u_{\lambda}~{}\mbox{ weakly in
}\mathcal{H}_{\lambda},\\\ u_{k}\to u_{\lambda}~{}\mbox{ point-wisely in
}V,\\\ u_{k}\to u_{\lambda}~{}\mbox{ strongly in }L^{p}(V)\mbox{ for
}p\in[2,+\infty].\end{cases}$

Thus, together with the weak-lower semi-continuity of norm and Fatou’s lemma,
we get

$\displaystyle\begin{aligned}
&\int_{V}\left(\Gamma(u_{\lambda}^{+})+\left(\lambda
a(x)+1\right)|u_{\lambda}^{+}|^{2}\right)d\mu-\int_{V}(|u_{\lambda}^{+}|^{2}\log|u_{\lambda}^{+}|^{2})^{-}d\mu-\frac{1}{2}K_{V}^{\lambda}(u)\\\
\leq&\liminf_{k\to+\infty}\left[\int_{V}\left(\Gamma(u_{k}^{+})+\left(\lambda
a(x)+1\right)|u_{k}^{+}|^{2}\right)d\mu-\int_{V}(|u_{k}^{+}|^{2}\log|u_{k}^{+}|^{2})^{-}d\mu-\frac{1}{2}K_{V}^{k}(u)\right]\\\
=&\liminf_{k\to+\infty}\int_{V}\left(|u_{k}^{+}|^{2}+(|u_{k}^{+}|^{2}\log|u_{k}^{+}|^{2})^{+}\right)d\mu\\\
=&\int_{V}|u_{\lambda}^{+}|^{2}d\mu+\int_{V}(|u_{\lambda}^{+}|^{2}\log|u_{\lambda}^{+}|^{2})^{+}d\mu,\end{aligned}$

where $K_{V}^{\lambda}(u)=\sum\limits_{x\in V}\sum\limits_{y\sim
x}\left[u_{\lambda}^{+}(x)u_{\lambda}^{-}(y)+u_{\lambda}^{-}(x)u_{\lambda}^{+}(y)\right]$.
It follows that

$J^{\prime}_{\lambda}(u_{\lambda})\cdot
u_{\lambda}^{+}=\int_{V}\left(\Gamma(u_{\lambda}^{+})+\lambda
a(x)|u_{\lambda}^{+}|^{2}\right)d\mu-\int_{V}|u_{\lambda}^{+}|^{2}\log|u_{\lambda}^{+}|^{2}d\mu-\frac{1}{2}K_{V}^{\lambda}(u)\leq
0.$

Similarly, it holds that

$J^{\prime}_{\lambda}(u_{\lambda})\cdot
u_{\lambda}^{-}=\int_{V}\left(\Gamma(u_{\lambda}^{-})+\lambda
a(x)|u_{\lambda}^{-}|^{2}\right)d\mu-\int_{V}|u_{\lambda}^{-}|^{2}\log|u_{\lambda}^{-}|^{2}d\mu-\frac{1}{2}K_{V}^{\lambda}(u)\leq
0.$

In view of Lemma 3.2 and Lemma 3.3, there exist $s,t\in(0,1]$ such that
$\widetilde{u}=su_{\lambda}^{+}+tu_{\lambda}^{-}\in\mathcal{M}_{\lambda}$.
Then

$\displaystyle\begin{aligned}
m_{\lambda}\leq&J_{\lambda}(\widetilde{u})=J_{\lambda}(\widetilde{u})-\frac{1}{2}J^{\prime}_{\lambda}(\widetilde{u})\cdot(su_{\lambda}^{+})-\frac{1}{2}J^{\prime}_{\lambda}(\widetilde{u})\cdot(tu_{\lambda}^{-})\\\
=&\frac{s^{2}}{2}\|u_{\lambda}^{+}\|_{2}^{2}+\frac{t^{2}}{2}\|u_{\lambda}^{-}\|_{2}^{2}\\\
\leq&\liminf_{k\to+\infty}\left[\frac{1}{2}\|u_{k}^{+}\|_{2}^{2}+\frac{1}{2}\|u_{k}^{-}\|_{2}^{2}\right]\\\
=&\liminf_{k\to+\infty}\left[J_{\lambda}(u_{k})-\frac{1}{2}J^{\prime}_{\lambda}(u_{k})\cdot
u_{k}^{+}-\frac{1}{2}J^{\prime}_{\lambda}(u_{k})\cdot u_{k}^{-}\right]\\\
=&\liminf_{k\to+\infty}J_{\lambda}(u_{k})=m_{\lambda}.\end{aligned}$

This implies that $s=t=1$, i.e., $u_{\lambda}\in\mathcal{M}_{\lambda}$
satisfying $J_{\lambda}(u_{\lambda})=m_{\lambda}$.

We claim that $m_{\lambda}>0$. In fact, if $m_{\lambda}=0$, we have

$0=J_{\lambda}(u_{\lambda})-\frac{1}{2}J^{\prime}(u_{\lambda})\cdot
u_{\lambda}^{+}-\frac{1}{2}J^{\prime}(u_{\lambda})\cdot
u_{\lambda}^{-}=\frac{1}{2}\|u_{\lambda}^{+}\|_{2}^{2}+\frac{1}{2}\|u_{\lambda}^{-}\|_{2}^{2}.$

Then, by similar arguments as in (3.7), it follows that
$\|u_{\lambda}^{\pm}\|_{\mathcal{H}_{\lambda}}=0$. However, by Lemma 2.5, for
any $q>2$, there exists $C_{q}>0$ such that

$\|u_{\lambda}^{\pm}\|_{\mathcal{H}_{\lambda}}^{2}<\int_{V}|u_{\lambda}^{\pm}|^{2}\log|u_{\lambda}^{\pm}|^{2}d\mu\leq\int_{V}(|u_{\lambda}^{\pm}|^{2}\log|u_{\lambda}^{\pm}|^{2})^{+}d\mu\leq
C_{q}\int_{V}|u_{\lambda}^{\pm}|^{q}d\mu\leq
C\|u_{\lambda}^{\pm}\|_{\mathcal{H}_{\lambda}}^{q},$

which provides a contradiction. Hence the claim holds. ∎

The following lemma completes the proof of Theorem 1.1.

######  Lemma 3.7.

If $u\in\mathcal{M}_{\lambda}$ with $J_{\lambda}(u)=m_{\lambda}$, then $u$ is
a sign-changing solution of equation (1.2). Moreover,
$m_{\lambda}>2c_{\lambda}$.

###### Proof.

We assume by contradiction that $u\in\mathcal{M}_{\lambda}$ with
$J_{\lambda}(u)=m_{\lambda}$, but $u$ is not a solution of equation (1.2).
Then we can find a function $\phi\in C_{c}(V)$ such that

$\int_{V}\left(\nabla u\nabla\phi+\lambda
a(x)u\phi\right)d\mu-\int_{V}u\phi\log u^{2}d\mu\leq-1,$

which implies that, for some $\varepsilon>0$ small enough,

$J_{\lambda}^{\prime}(su^{+}+tu^{-}+\sigma\phi)\cdot\phi\leq-\frac{1}{2}\
\mbox{for all }|s-1|+|t-1|+|\sigma|\leq\epsilon.$

In what follows, we estimate
$\sup\limits_{s,t}J_{\lambda}\left(su^{+}+tu^{-}+\varepsilon\eta(s,t)\phi\right)$,
where $\eta$ is a cut-off function such that

$\eta(s,t)=\begin{cases}1~{}~{}~{}\mbox{if
}|s-1|\leq\frac{1}{2}\varepsilon\mbox{ and
}|t-1|\leq\frac{1}{2}\varepsilon,\\\ 0~{}~{}~{}\mbox{if
}|s-1|\geq\varepsilon\mbox{ or }|t-1|\geq\varepsilon.\end{cases}$

In the case of $|s-1|\leq\varepsilon$ and $|t-1|\leq\varepsilon$, we have

$\displaystyle
J_{\lambda}\left(su^{+}+tu^{-}+\varepsilon\eta(s,t)\phi\right)=$
$\displaystyle
J_{\lambda}\left(su^{+}+tu^{-}+\varepsilon\eta(s,t)\phi\right)-J_{\lambda}(su^{+}+tu^{-})+J_{\lambda}(su^{+}+tu^{-})$
$\displaystyle=$ $\displaystyle
J_{\lambda}(su^{+}+tu^{-})+\int_{0}^{1}J_{\lambda}^{\prime}\left(su^{+}+tu^{-}+\sigma\varepsilon\eta(s,t)\phi\right)\cdot\left(\varepsilon\eta(s,t)\phi\right)d\sigma$
$\displaystyle=$ $\displaystyle
J_{\lambda}(su^{+}+tu^{-})+\varepsilon\eta(s,t)\int_{0}^{1}J_{\lambda}^{\prime}\left(su^{+}+tu^{-}+\sigma\varepsilon\eta(s,t)\phi\right)\cdot\phi
d\sigma$ $\displaystyle\leq$ $\displaystyle
J_{\lambda}(su^{+}+tu^{-})-\frac{1}{2}\varepsilon\eta(s,t).$

For the other case, that is $|s-1|\geq\varepsilon$ or $|t-1|\geq\varepsilon$,
$\eta(s,t)=0$, the above estimate is obvious. Now since
$u\in\mathcal{M}_{\lambda}$, for $(s,t)\neq(1,1)$, by Lemma 3.1, we have
$J_{\lambda}(su^{+}+tu^{-})<J_{\lambda}(u)$. Hence

$J_{\lambda}\left(su^{+}+tu^{-}+\varepsilon\eta(s,t)\phi\right)\leq
J_{\lambda}(su^{+}+tu^{-})<J_{\lambda}(u)\ \mbox{for all }(s,t)\neq(1,1).$

For $(s,t)=(1,1)$,

$J_{\lambda}\left(su^{+}+tu^{-}+\varepsilon\eta(s,t)\phi\right)\leq
J_{\lambda}(su^{+}+tu^{-})-\frac{1}{2}\varepsilon\eta(1,1)=J_{\lambda}(u)-\frac{1}{2}\varepsilon.$

In any case, we have
$J_{\lambda}\left(su^{+}+tu^{-}+\varepsilon\eta(s,t)\phi\right)<J_{\lambda}(u)=m_{\lambda}$.
In particular, for $0<\varepsilon<1-\varepsilon$,

$\sup\limits_{\varepsilon\leq s,t\leq
2-\varepsilon}J_{\lambda}\left(su^{+}+tu^{-}+\varepsilon\eta(s,t)\phi\right)=\widetilde{m}_{\lambda}<m_{\lambda}.$

Set $v=su^{+}+tu^{-}+\varepsilon\eta(s,t)\phi$ and define

$H(s,t)=\left(F_{1}(s,t),F_{2}(s,t)\right)\dot{=}\left(J_{\lambda}^{\prime}(v)\cdot
v^{+},J_{\lambda}^{\prime}(v)\cdot v^{-}\right).$

By the definition of $\eta$, when $s=\varepsilon,\ t\in(\epsilon,2-\epsilon)$,
we have $\eta(s,t)=0$ and $s<t$. Hence

$\displaystyle F_{1}(\varepsilon,t)\dot{=}$ $\displaystyle
J^{\prime}_{\lambda}(su^{+}+tu^{-})\cdot(su^{+})\Big{|}_{s=\varepsilon}$
$\displaystyle=$ $\displaystyle\left[s^{2}J^{\prime}_{\lambda}(u^{+})\cdot
u^{+}-\frac{st}{2}K_{V}(u)-s^{2}\log
s^{2}\|u^{+}\|_{2}^{2}\right]_{s=\varepsilon}$ $\displaystyle>$
$\displaystyle\left[s^{2}\left(J^{\prime}_{\lambda}(u^{+})-\frac{1}{2}K_{V}(u)\right)-s^{2}\log
s^{2}\|u^{+}\|_{2}^{2}\right]_{s=\varepsilon}$ $\displaystyle=$
$\displaystyle-\varepsilon^{2}\log\varepsilon^{2}\|u^{+}\|_{2}^{2}$
$\displaystyle>$ $\displaystyle 0.$

When $s=2-\varepsilon,\ t\in(\epsilon,2-\epsilon)$, we have $\eta(s,t)=0$ and
$s>t$. Therefore,

$\displaystyle F_{1}(2-\varepsilon,t)\dot{=}$ $\displaystyle
J^{\prime}_{\lambda}(su^{+}+tu^{-})\cdot(su^{+})\Big{|}_{s=2-\varepsilon}$
$\displaystyle=$ $\displaystyle\left[s^{2}J^{\prime}_{\lambda}(u^{+})\cdot
u^{+}-\frac{st}{2}K_{V}(u)-s^{2}\log
s^{2}\|u^{+}\|_{2}^{2}\right]_{s=2-\varepsilon}$ $\displaystyle<$
$\displaystyle\left[s^{2}\left(J^{\prime}_{\lambda}(u^{+})-\frac{1}{2}K_{V}(u)\right)-s^{2}\log
s^{2}\|u^{+}\|_{2}^{2}\right]_{s=2-\varepsilon}$ $\displaystyle=$
$\displaystyle-(2-\varepsilon)^{2}\log(2-\varepsilon)^{2}\|u^{+}\|_{2}^{2}$
$\displaystyle<$ $\displaystyle 0.$

That is

$\displaystyle F_{1}(\varepsilon,t)>0,\ F_{1}(2-\varepsilon,t)<0\ \mbox{for
all }t\in(\varepsilon,2-\varepsilon).$

Similarly, we have

$\displaystyle F_{2}(s,\varepsilon)>0,\ F_{2}(s,2-\varepsilon)<0\ \mbox{for
all }s\in(\varepsilon,2-\varepsilon).$

Thus, applying Miranda’s theorem [40], there exists
$(s_{0},t_{0})\in(\varepsilon,2-\varepsilon)\times(\varepsilon,2-\varepsilon)$
such that
$\widetilde{u}=s_{0}u^{+}+t_{0}u^{-}+\varepsilon\eta(s_{0},t_{0})\phi\in\mathcal{M}_{\lambda}$
and $J_{\lambda}(\widetilde{u})<m_{\lambda}$. This give a contradiction to the
definition of $m_{\lambda}$.

Next, we prove that $m_{\lambda}>2c_{\lambda}$. Assume that
$u\in\mathcal{M}_{\lambda}$ such that $J_{\lambda}(u)=m_{\lambda}$. Then
$u^{\pm}\neq 0$. Similar to the proof of Lemma 3.2 and Lemma 3.3, we can
deduce that there exists a unique $s_{u^{+}}\in(0,1]$ such that
$s_{u^{+}}u^{+}\in\mathcal{N}_{\lambda}$, and a unique $t_{u^{-}}\in(0,1]$
such that $t_{u^{-}}u^{-}\in\mathcal{N}_{\lambda}$. Similar to the proofs of
Lemma 3.6 and Lemma 3.7, we can deduce that $c_{\lambda}>0$ can be achieved.
Furthermore, if $u\in\mathcal{N}_{\lambda}$ with $J_{\lambda}(u)=c_{\lambda}$,
then $u$ is a least energy solution.

By the definition of $J_{\lambda}$ and $K_{V}(x,y)<0$, we have

$\displaystyle J_{\lambda}(s_{u^{+}}u^{+}+t_{u^{-}}u^{-})=$ $\displaystyle
J_{\lambda}(s_{u^{+}}u^{+})+J_{\lambda}(t_{u^{-}}u^{-})-\frac{s_{u^{+}}t_{u^{-}}}{2}K_{V}(u)$
$\displaystyle>$ $\displaystyle
J_{\lambda}(s_{u^{+}}u^{+})+J_{\lambda}(t_{u^{-}}u^{-}).$

By Lemma 3.1, we deduce that

$\displaystyle m_{\lambda}=J_{\lambda}(u^{+}+u^{-})\geq
J_{\lambda}(s_{u^{+}}u^{+}+t_{u^{-}}u^{-})>J_{\lambda}(s_{u^{+}}u^{+})+J_{\lambda}(t_{u^{-}}u^{-})\geq
2c_{\lambda}.$

This completes the proof. ∎

###### Proof of Theorem 1.2.

The proof can be obtained by similar arguments in Theorem 1.1. ∎

##  4 Convergence of the least energy sign-changing solution $u_{\lambda}$

In this section, we shall study the asymptotic behavior of $u_{\lambda}$ as
$\lambda\to+\infty$. Firstly, we show that $\\{u_{\lambda}\\}$ is uniformly
bounded above and below away from zero.

######  Lemma 4.1.

There exists $\sigma>0$ (independent of $\lambda$) such that
$\|u\|_{\mathcal{H}_{\lambda}}\geq\|u\|_{H^{1}(V)}\geq\sigma$ for all
$u\in\mathcal{M}_{\lambda}$.

###### Proof.

Note that for all $\varepsilon>0$, if $s\geq e^{-\frac{1}{2}}$, then

$e^{\frac{\varepsilon}{2}}s^{2+\varepsilon}\geq s^{2}.$ (4.1)

Since $u\in\mathcal{M}_{\lambda}$, by Lemma 2.5 and (4.1), we have

$\displaystyle 0=$ $\displaystyle J^{\prime}_{\lambda}(u)\cdot
u^{+}=J^{\prime}_{\lambda}(u^{+})\cdot u^{+}-\frac{1}{2}K_{V}(u)$
$\displaystyle\geq$ $\displaystyle\int_{V}\left(\Gamma(u^{+})+(\lambda
a(x)+1)|u^{+}|^{2}\right)d\mu-\int_{V}|u^{+}|^{2}d\mu-\int_{V}|u^{+}|^{2}\log|u^{+}|^{2}d\mu$
$\displaystyle=$
$\displaystyle\|u^{+}\|_{\mathcal{H}_{\lambda}}^{2}-\int_{|u^{+}|<e^{-\frac{1}{2}}}\left(|u^{+}|^{2}+|u^{+}|^{2}\log|u^{+}|^{2}\right)d\mu-\int_{|u^{+}|\geq
e^{-\frac{1}{2}}}|u^{+}|^{2}d\mu$
$\displaystyle-\int_{e^{-\frac{1}{2}}\leq|u^{+}|\leq
1}|u^{+}|^{2}\log|u^{+}|^{2}d\mu-\int_{|u^{+}|>1}|u^{+}|^{2}\log|u^{+}|^{2}d\mu$
$\displaystyle\geq$
$\displaystyle\|u^{+}\|_{\mathcal{H}_{\lambda}}^{2}-e^{\frac{\varepsilon}{2}}\int_{|u^{+}|\geq
e^{-\frac{1}{2}}}|u^{+}|^{2+\varepsilon}d\mu-
C_{\varepsilon}\int_{|u^{+}|>1}|u^{+}|^{2+\varepsilon}d\mu$
$\displaystyle\geq$
$\displaystyle\|u^{+}\|_{\mathcal{H}_{\lambda}}^{2}-C_{\varepsilon}^{\prime}\int_{V}|u^{+}|^{2+\varepsilon}d\mu$
$\displaystyle\geq$
$\displaystyle\|u^{+}\|_{H^{1}(V)}^{2}-C_{\varepsilon}^{\prime\prime}\|u^{+}\|_{H^{1}(V)}^{2+\varepsilon}.$

Then

$\|u^{+}\|_{\mathcal{H}_{\lambda}}\geq\|u^{+}\|_{H^{1}(V)}\geq(C^{\prime\prime}_{\varepsilon})^{-\frac{1}{\varepsilon}}>0.$

Similarly, we get

$\|u^{-}\|_{\mathcal{H}_{\lambda}}\geq\|u^{-}\|_{H^{1}(V)}\geq(C^{\prime\prime}_{\varepsilon})^{-\frac{1}{\varepsilon}}>0.$

Hence,

$\displaystyle\|u\|_{\mathcal{H}_{\lambda}}^{2}\geq\|u\|_{H^{1}(V)}^{2}=\|u^{+}\|_{H^{1}(V)}^{2}+\|u^{-}\|_{H^{1}(V)}^{2}-K_{V}(u)>\|u^{+}\|_{H^{1}(V)}^{2}+\|u^{-}\|_{H^{1}(V)}^{2}\geq
2(C^{\prime\prime}_{\varepsilon})^{-\frac{2}{\varepsilon}}.$

Thus we can choose
$\sigma=\sqrt{2}(C^{\prime\prime}_{\varepsilon})^{-\frac{1}{\varepsilon}}$
such that $\|u\|_{\mathcal{H}_{\lambda}}\geq\|u\|_{H^{1}(V)}\geq\sigma$. ∎

######  Lemma 4.2.

There exists $c_{0}>0$ (independent of $\lambda$) such that if sequence
$\\{u_{k}\\}\subset\mathcal{M}_{\lambda}$ of $J_{\lambda}$ with
$\lim_{k\to\infty}J_{\lambda}(u_{k})=m_{\lambda}$, then
$\|u_{k}\|_{\mathcal{H}_{\lambda}}\leq c_{0}$.

###### Proof.

Since $\mathcal{M}_{\Omega}\subset\mathcal{M}_{\lambda}$, it is easily seen
that $m_{\lambda}\leq m_{\Omega}$ for any $\lambda>0$. Since
$\\{u_{k}\\}\subset\mathcal{M}_{\lambda}$ and
$\lim_{k\to\infty}J_{\lambda}(u_{k})=m_{\lambda}$, we have

$\displaystyle\lim_{k\to+\infty}J_{\lambda}(u_{k})=$
$\displaystyle\lim_{k\to+\infty}\left[J_{\lambda}(u_{k})-\frac{1}{2}J^{\prime}_{\lambda}(u_{k})\cdot
u_{k}^{+}-\frac{1}{2}J_{\lambda}^{\prime}(u_{k})\cdot u_{k}^{-}\right]$ (4.2)
$\displaystyle=$
$\displaystyle\lim_{k\to+\infty}\left(\frac{1}{2}\|u_{k}^{+}\|_{2}^{2}+\frac{1}{2}\|u_{k}^{-}\|_{2}^{2}\right)=m_{\lambda}\leq
m_{\Omega}.$

By Lemma 2.5, the Hölder’s inequality and Young inequality, for any
$\varepsilon\in(0,1)$, there exist
$C_{\varepsilon},C_{\varepsilon}^{\prime},C_{\varepsilon}^{\prime\prime}>0$
such that

$\displaystyle\int_{V}|u_{k}^{\pm}|^{2}\log|u_{k}^{\pm}|^{2}d\mu\leq$
$\displaystyle\int_{V}(|u_{k}^{\pm}|^{2}\log|u_{k}^{\pm}|^{2})^{+}d\mu\leq
C_{\varepsilon}\int_{V}|u_{k}^{\pm}|^{2+\varepsilon}d\mu$ $\displaystyle\leq$
$\displaystyle
C_{\varepsilon}\left(\int_{V}|u_{k}^{\pm}|^{2}d\mu\right)^{\frac{1}{2}}\left(\int_{V}|u_{k}^{\pm}|^{2(1+\varepsilon)}d\mu\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C_{\varepsilon}^{\prime}\|u_{k}^{\pm}\|_{2}\|u_{k}^{\pm}\|_{\mathcal{H}_{\lambda}}^{1+\varepsilon}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\|u_{k}^{\pm}\|_{\mathcal{H}_{\lambda}}^{2}+C_{\varepsilon}^{\prime\prime}\|u_{k}^{\pm}\|_{2}^{\frac{2}{1-\varepsilon}}.$

Since $\\{u_{k}\\}\subset\mathcal{M}_{\lambda}$, we deduce that

$\displaystyle\|u_{k}^{\pm}\|_{\mathcal{H}_{\lambda}}^{2}-\frac{1}{2}K_{V}^{k}(x,y)\leq\frac{1}{2}\|u_{k}^{\pm}\|_{\mathcal{H}_{\lambda}}^{2}+C_{\varepsilon}^{\prime\prime}\|u_{k}^{\pm}\|_{2}^{\frac{2}{1-\varepsilon}}+\|u_{k}^{\pm}\|_{2}^{2}.$

This together with (4.2) implies

$\displaystyle\lim_{k\to+\infty}\left(\|u_{k}^{\pm}\|_{\mathcal{H}_{\lambda}}^{2}-\frac{1}{2}K_{V}^{k}(u)\right)\leq
C_{\varepsilon}^{\prime\prime\prime}\left(m_{\Omega}^{\frac{1}{1-\varepsilon}}+m_{\Omega}\right).$

From Lemma 3.6 we know $m_{\lambda}>0$ and $m_{\Omega}>0$. Therefore it
suffices to choose
$c_{0}=C_{\varepsilon}^{\prime\prime\prime}\left(m_{\Omega}^{\frac{1}{1-\varepsilon}}+m_{\Omega}\right)$.
∎

Secondly, we have the following relation about $m_{\lambda}$ and $m_{\Omega}$.

######  Lemma 4.3.

$m_{\lambda}\to m_{\Omega}$ as $\lambda\to+\infty$.

###### Proof.

By $m_{\lambda}\leq m_{\Omega}$ for any $\lambda>0$, passing to subsequence if
necessary, we may take a sequence $\lambda_{k}\to+\infty$ such that

$\lim_{k\to\infty}m_{\lambda_{k}}=\eta\leq m_{\Omega},$ (4.3)

where
$m_{\lambda_{k}}=\inf\limits_{u_{k}\in\mathcal{M}_{\lambda_{k}}}J_{\lambda_{k}}(u_{k})$.
Then, combining Lemma 4.1 and (3.7), it is easy to get $\eta>0$. By Lemma 4.2,
we have that $\\{u_{\lambda_{k}}\\}$ is uniformly bounded in
$\mathcal{H}_{\lambda_{k}}$. Consequently, $\\{u_{\lambda_{k}}\\}$ is also
bounded in $H^{1}(V)$ and thus, up to a subsequence, there exists some
$u_{0}\in H^{1}(V)$ such that

$\begin{cases}u_{\lambda_{k}}\rightharpoonup u_{0}~{}\mbox{ weakly in
}H^{1}(V),\\\ u_{\lambda_{k}}\to u_{0}~{}\mbox{ point-wisely in }V,\\\
u_{\lambda_{k}}\to u_{0}~{}\mbox{ strongly in }L^{p}(V)\mbox{ for
}p\in[2,+\infty].\end{cases}$ (4.4)

We claim that $u_{0}\mid_{\Omega^{c}}=0$. In fact, if there exists a vertex
$x_{0}\in\Omega^{c}$ such that $u_{0}(x_{0})\neq 0$. Since
$u_{\lambda_{k}}\in\mathcal{M}_{\lambda_{k}}$, we have

$\displaystyle J_{\lambda_{k}}(u_{\lambda_{k}})$
$\displaystyle=\frac{1}{2}\|u_{\lambda_{k}}\|_{\mathcal{H}_{\lambda_{k}}}^{2}-\frac{1}{2}\int_{V}u_{\lambda_{k}}^{2}\log
u_{\lambda_{k}}^{2}d\mu$
$\displaystyle\geq\frac{\lambda_{k}}{2}\int_{V}a(x)u_{\lambda_{k}}^{2}d\mu-\frac{1}{2}\int_{V}(u_{\lambda_{k}}^{2}\log
u_{\lambda_{k}}^{2})^{+}d\mu$
$\displaystyle\geq\frac{\lambda_{k}}{2}\int_{V}a(x)u_{\lambda_{k}}^{2}d\mu-\frac{C_{\varepsilon}}{2}\int_{V}|u_{\lambda_{k}}|^{2+\varepsilon}d\mu$
$\displaystyle\geq\frac{\lambda_{k}}{2}\sum_{x\in
V}\mu(x)a(x)u^{2}_{\lambda_{k}}(x)-C^{\prime}_{\varepsilon}\|u_{\lambda_{k}}\|_{H^{1}(V)}^{2+\varepsilon}$
$\displaystyle\geq\frac{\lambda_{k}}{2}\mu_{\min}a(x_{0})u^{2}_{\lambda_{k}}(x_{0})-C^{\prime\prime}_{\varepsilon}.$

Since $a(x_{0})>0$, $u_{\lambda_{k}}(x_{0})\to u_{0}(x_{0})\neq 0$ and
$\lambda_{k}\to+\infty$, we get

$\lim_{k\to+\infty}J_{\lambda_{k}}(u_{\lambda_{k}})=+\infty,$

This is in contradiction with (4.3). Hence the claim holds.

Since $u_{0}\mid_{\Omega^{c}}=0$, by the weak lower semi-continuity of the
norm $\|\cdot\|_{H^{1}(V)}$ and Fatou’s lemma, taking $u_{\lambda_{k}}^{+}$ as
test function in equation (1.2), we get

$\displaystyle\int_{\Omega\cup\partial\Omega}\Gamma(u_{0}^{+})d\mu+\int_{\Omega}|u_{0}^{+}|^{2}d\mu-\int_{\\{\Omega:|u_{0}^{+}|\leq
1\\}}|u_{0}^{+}|^{2}\log|u_{0}^{+}|^{2}d\mu-\frac{1}{2}K_{\Omega}^{0}(u)$
$\displaystyle\leq$
$\displaystyle\int_{V}\left(\Gamma(u_{0}^{+})+|u_{0}^{+}|^{2}\right)d\mu-\int_{\\{V:|u_{0}^{+}|\leq
1\\}}|u_{0}^{+}|^{2}\log|u_{0}^{+}|^{2}d\mu-\frac{1}{2}K_{V}^{0}(u)$
$\displaystyle\leq$
$\displaystyle\liminf_{k\to+\infty}\left[\int_{V}\left(\Gamma(u_{\lambda_{k}}^{+})+\left(\lambda_{k}a(x)+1\right)|u_{\lambda_{k}}^{+}|^{2}\right)d\mu-\int_{\\{V:|u_{\lambda_{k}}^{+}|\leq
1\\}}|u_{\lambda_{k}}^{+}|^{2}\log|u_{\lambda_{k}}^{+}|^{2}d\mu-\frac{1}{2}K_{V}^{\lambda_{k}}(u)\right]$
$\displaystyle=$
$\displaystyle\liminf_{k\to+\infty}\left[\int_{V}|u_{\lambda_{k}}^{+}|^{2}d\mu+\int_{\\{V:|u_{\lambda_{k}}^{+}|>1\\}}|u_{\lambda_{k}}^{+}|^{2}\log|u_{\lambda_{k}}^{+}|^{2}d\mu\right]$
$\displaystyle=$
$\displaystyle\int_{V}|u_{0}^{+}|^{2}d\mu+\int_{\\{V:|u_{0}^{+}|>1\\}}|u_{0}^{+}|^{2}\log|u_{0}^{+}|^{2}d\mu$
$\displaystyle=$
$\displaystyle\int_{\Omega}|u_{0}^{+}|^{2}d\mu+\int_{\\{\Omega:|u_{0}^{+}|>1\\}}|u_{0}^{+}|^{2}\log|u_{0}^{+}|^{2}d\mu,$

where

$\displaystyle
K_{\Omega}^{0}(u)=\sum_{x\in\Omega\cup\partial\Omega}\sum_{y\sim
x}\left[u_{0}^{+}(x)u_{0}^{-}(y)+u_{0}^{-}(x)u_{0}^{+}(y)\right],$
$\displaystyle K_{V}^{0}(u)=\sum_{x\in V}\sum_{y\sim
x}\left[u_{0}^{+}(x)u_{0}^{-}(y)+u_{0}^{-}(x)u_{0}^{+}(y)\right],$
$\displaystyle K_{V}^{\lambda_{k}}(u)=\sum_{x\in V}\sum_{y\sim
x}\left[u_{\lambda_{k}}^{+}(x)u_{\lambda_{k}}^{-}(y)+u_{\lambda_{k}}^{-}(x)u_{\lambda_{k}}^{+}(y)\right].$

Then

$J^{\prime}_{\Omega}(u_{0})\cdot
u_{0}^{+}=\int_{\Omega\cup\partial\Omega}\Gamma(u_{0}^{+})d\mu-\int_{\Omega}|u_{0}^{+}|^{2}\log|u_{0}^{+}|^{2}d\mu-\frac{1}{2}K_{\Omega}^{0}(u)\leq
0.$

Similarly, it holds that

$J^{\prime}_{\Omega}(u_{0})\cdot
u_{0}^{-}=\int_{\Omega\cup\partial\Omega}\Gamma(u_{0}^{-})d\mu-\int_{\Omega}|u_{0}^{-}|^{2}\log|u_{0}^{-}|^{2}d\mu-\frac{1}{2}K_{\Omega}^{0}(u)\leq
0.$

In view of Lemma 3.4 and Lemma 3.5, there exist $s,t\in(0,1]$ such that
$\widetilde{u}_{0}=su_{0}^{+}+tu_{0}^{-}\in\mathcal{M}_{\Omega}$. Then

$\displaystyle m_{\Omega}\leq$ $\displaystyle
J_{\Omega}(\widetilde{u}_{0})=J_{\Omega}(\widetilde{u}_{0})-\frac{1}{2}J^{\prime}_{\Omega}(\widetilde{u}_{0})\cdot(su_{0}^{+})-\frac{1}{2}J^{\prime}_{\Omega}(\widetilde{u}_{0})\cdot(tu_{0}^{-})$
$\displaystyle=$
$\displaystyle\frac{s^{2}}{2}\|u_{0}^{+}\|_{L^{2}(\Omega)}^{2}+\frac{t^{2}}{2}\|u_{0}^{-}\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq$
$\displaystyle\liminf_{k\to\infty}\left[\frac{1}{2}\|u_{\lambda_{k}}^{+}\|_{2}^{2}+\frac{1}{2}\|u_{\lambda_{k}}^{-}\|_{2}^{2}\right]$
$\displaystyle=$
$\displaystyle\liminf_{k\to\infty}\left[J_{\lambda_{k}}(u_{\lambda_{k}})-\frac{1}{2}J^{\prime}_{\lambda_{k}}(u_{\lambda_{k}})\cdot
u_{\lambda_{k}}^{+}-\frac{1}{2}J^{\prime}_{\lambda_{k}}(u_{\lambda_{k}})\cdot
u_{\lambda_{k}}^{-}\right]$ $\displaystyle=$
$\displaystyle\liminf_{k\to+\infty}J_{\lambda_{k}}(u_{\lambda_{k}})=\eta\leq
m_{\Omega}.$

Hence, $\lim_{\lambda\to+\infty}m_{\lambda}=m_{\Omega}$. This completes the
proof. ∎

###### Proof of Theorem 1.3.

Assume that $u_{\lambda_{k}}\in\mathcal{M}_{\lambda_{k}}$ satisfies
$J_{\lambda_{k}}(u_{\lambda_{k}})=m_{\lambda_{k}}$. We shall prove that
$u_{\lambda_{k}}$ converges in $H^{1}(V)$ to a least energy sign-changing
solution $u_{0}$ of equation $(\ref{eq-eq2})$ along a subsequence.

Lemma 4.2 gives that $u_{\lambda_{k}}\in\mathcal{H}_{\lambda_{k}}$ is
uniformly bounded. Consequently, we have that $\\{u_{\lambda_{k}}\\}$ is also
bounded in $H^{1}(V)$. Therefore, we can assume that for any $p\in[2,\infty)$,
$u_{\lambda_{k}}\to u_{0}$ in $L^{p}(V)$ and $u_{\lambda_{k}}\rightharpoonup
u_{0}$ in $H^{1}(V)$. Moreover, in view of $u_{0}\in\mathcal{N}_{\Omega}$ and
we get from Lemma 4.1 that $u_{0}\not\equiv 0$. As proved in Lemma 4.3, we can
prove that $u_{0}\mid_{\Omega^{c}}=0$. Then it suffices to show that, as
$k\to+\infty$, we have
$\lambda_{k}\int_{V}a(x)|u_{\lambda_{k}}^{\pm}|^{2}d\mu\to 0$ and
$\int_{V}\Gamma(u_{\lambda_{k}}^{\pm})d\mu\to\int_{V}\Gamma(u_{0}^{\pm})d\mu$.
If not, we may assume that

$\lim\limits_{k\to+\infty}\lambda_{k}\int_{V}a(x)|u_{\lambda_{k}}^{\pm}|^{2}d\mu=\delta>0.$

Since $u_{0}\mid_{\Omega^{c}}=0$, by weak lower semi-continuity of the norm
$\|\cdot\|_{H^{1}(V)}$ and Fatou’s lemma, taking $u_{\lambda_{k}}^{+}$ as test
function in equation (1.2), we get

$\displaystyle\int_{\Omega\cup\partial\Omega}\Gamma(u_{0}^{+})d\mu+\int_{\Omega}|u_{0}^{+}|^{2}d\mu-\int_{\\{\Omega:|u_{0}^{+}|\leq
1\\}}|u_{0}^{+}|^{2}\log|u_{0}^{+}|^{2}d\mu-\frac{1}{2}K_{\Omega}^{0}(u)$
$\displaystyle<$
$\displaystyle\int_{V}\left(\Gamma(u_{0}^{+})+|u_{0}^{+}|^{2}\right)d\mu+\delta-\int_{\\{V:|u_{0}^{+}|\leq
1\\}}|u_{0}^{+}|^{2}\log|u_{0}^{+}|^{2}d\mu-\frac{1}{2}K_{V}^{0}(u)$
$\displaystyle\leq$
$\displaystyle\liminf_{k\to+\infty}\left[\int_{V}\left(\Gamma(u_{\lambda_{k}}^{+})+\left(\lambda_{k}a(x)+1\right)|u_{\lambda_{k}}^{+}|^{2}\right)d\mu-\int_{\\{V:|u_{\lambda_{k}}^{+}|\leq
1\\}}|u_{\lambda_{k}}^{+}|^{2}\log|u_{\lambda_{k}}^{+}|^{2}d\mu-\frac{1}{2}K_{V}^{\lambda_{k}}(u)\right]$
$\displaystyle=$
$\displaystyle\liminf_{k\to+\infty}\left[\int_{V}|u_{\lambda_{k}}^{+}|^{2}d\mu+\int_{\\{V:|u_{\lambda_{k}}^{+}|>1\\}}|u_{\lambda_{k}}^{+}|^{2}\log|u_{\lambda_{k}}^{+}|^{2}d\mu\right]$
$\displaystyle=$
$\displaystyle\int_{V}|u_{0}^{+}|^{2}d\mu+\int_{\\{V:|u_{0}^{+}|>1\\}}|u_{0}^{+}|^{2}\log|u_{0}^{+}|^{2}d\mu$
$\displaystyle=$
$\displaystyle\int_{\Omega}|u_{0}^{+}|^{2}d\mu+\int_{\\{\Omega:|u_{0}^{+}|>1\\}}|u_{0}^{+}|^{2}\log|u_{0}^{+}|^{2}d\mu,$

which implies that

$J^{\prime}_{\Omega}(u_{0})\cdot
u_{0}^{+}=\int_{\Omega\cup\partial\Omega}\Gamma(u_{0}^{+})d\mu-\int_{\Omega}|u_{0}^{+}|^{2}\log|u_{0}^{+}|^{2}d\mu-\frac{1}{2}K_{\Omega}^{0}(u)<0.$
(4.5)

Similarly, it holds that

$J^{\prime}_{\Omega}(u_{0})\cdot
u_{0}^{-}=\int_{\Omega\cup\partial\Omega}\Gamma(u_{0}^{-})d\mu-\int_{\Omega}|u_{0}^{-}|^{2}\log|u_{0}^{-}|^{2}d\mu-\frac{1}{2}K_{\Omega}^{0}(u)<0.$
(4.6)

By similar arguments as above, if

$\lim\limits_{k\to+\infty}\int_{V}\Gamma(u_{\lambda_{k}}^{\pm})d\mu>\int_{V}\Gamma(u_{0}^{\pm})d\mu,$

we also have (4.5) and (4.6).

In view of Lemma 3.4 and Lemma 3.5, there exist two constants $s,t\in(0,1)$
such that $\widetilde{u}_{0}=su_{0}^{+}+tu_{0}^{-}\in\mathcal{M}_{\Omega}$.
Consequently, we have

$\displaystyle m_{\Omega}\leq$ $\displaystyle
J_{\Omega}(\widetilde{u}_{0})=J_{\Omega}(\widetilde{u}_{0})-\frac{1}{2}J^{\prime}_{\Omega}(\widetilde{u}_{0})\cdot(su_{0}^{+})-\frac{1}{2}J^{\prime}_{\Omega}(\widetilde{u}_{0})\cdot(tu_{0}^{-})$
$\displaystyle=$
$\displaystyle\frac{s^{2}}{2}\|u_{0}^{+}\|_{L^{2}(\Omega)}^{2}+\frac{t^{2}}{2}\|u_{0}^{-}\|_{L^{2}(\Omega)}^{2}$
$\displaystyle<$
$\displaystyle\frac{1}{2}\|u_{0}^{+}\|_{2}^{2}+\frac{1}{2}\|u_{0}^{-}\|_{2}^{2}$
$\displaystyle\leq$
$\displaystyle\liminf_{k\to+\infty}\left[\frac{1}{2}\|u_{\lambda_{k}}^{+}\|_{2}^{2}+\frac{1}{2}\|u_{\lambda_{k}}^{-}\|_{2}^{2}\right]$
$\displaystyle=$
$\displaystyle\liminf_{k\to+\infty}\left[J_{\lambda_{k}}(u_{\lambda_{k}})-\frac{1}{2}J^{\prime}_{\lambda_{k}}(u_{\lambda_{k}})\cdot
u_{\lambda_{k}}^{+}-\frac{1}{2}J^{\prime}_{\lambda_{k}}(u_{\lambda_{k}})\cdot
u_{\lambda_{k}}^{-}\right]$ $\displaystyle=$
$\displaystyle\liminf_{k\to+\infty}J_{\lambda_{k}}(u_{\lambda_{k}})$
$\displaystyle=$
$\displaystyle\liminf_{k\to+\infty}m_{\lambda_{k}}=m_{\Omega},$

which leads to a contradiction. Hence, we obtain that $u_{\lambda_{k}}\to
u_{0}$ in $H^{1}(V)$ and $u_{0}$ is a least energy sign-changing solution of
problem (1.8). ∎

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data availability

No data was used for the research described in the article.

## Acknowledgements

The research of Xiaojun Chang was supported by National Natural Science
Foundation of China (Grant No.11971095), while Duokui Yan was supported by
National Natural Science Foundation of China (Grant No.11871086).

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