# Exact solutions of nonlinear delay reaction-diffusion equations with
variable coefficients

M.O. Aibinu<EMAIL_ADDRESS>/<EMAIL_ADDRESS>S. C. Thakur
<EMAIL_ADDRESS>S. Moyo<EMAIL_ADDRESS>Institute for Systems Science & KZN
E-Skill CoLab, Durban University of Technology, Durban 4000, South Africa
DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-
MaSS), South Africa  National Institute for Theoretical and Computational
Sciences (NITheCS), South Africa  KZN E-Skill CoLab, Durban University of
Technology, Durban 4000, South Africa  Institute for Systems Science & Office
of the DVC Research, Innovation & Engagement
Milena Court, Durban University of Technology, Durban 4000, South Africa

###### Abstract

A modified method of functional constraints is used to construct the exact
solutions of nonlinear equations of reaction-diffusion type with delay and
which are associated with variable coefficients. This study considers a most
generalized form of nonlinear equations of reaction-diffusion type with delay
and which are nonlinear and associated with variable coefficients. A novel
technique is used in this study to obtain the exact solutions which are new
and are of the form of traveling-wave solutions. Arbitrary functions are
present in the solutions and they also contain free parameters, which make
them suitable for usage in solving certain modeling problems, testing
numerical and approximate analytical methods. The results of this study also
find applications in obtaining the exact solutions of other forms of partial
differential equations which are more complex. Specific examples of nonlinear
equations of reaction-diffusion type with delay are given and their exact
solutions are presented. Solutions of certain reaction-diffusion equations are
also displayed graphically.

###### keywords:

Reaction-diffusion, Time delay, Exact solutions, Differential equations.

###### MSC:

[2010] 35K55 , 35B10 , 35B40 , 35K57

††journal: Partial Differential Equations in Applied Mathematics

##  1 Introduction

The roles which Nonlinear Partial Differential Equations (NPDEs) play are
prominent in the description and analysis of real-life processes and phenomena
(See e.g., [1, 2]). Therefore, it is pivotal to seek for the ways of obtaining
exact solutions of NPDEs for a proper and accurate analysis. Several processes
and phenomena which occur in sciences and engineering lead to NPDEs as there
are several conditions and parameters to be considered in the modeling of such
systems. Reaction-Diffusion Equations (RDEs) are members of NPDEs. Reaction-
diffusion systems can be described as mathematical models which find
applications in diverse physical phenomena. In its simplest form in one
spatial dimension, the RDE has the form

$u_{t}=Du_{xx}+G(u),$ (1.1)

where $u(x,t)$ denotes the unknown function, $G$ accounts for all local
reactions, and $D$ is a diffusion coefficient (which is a constant) (See e.g,
[3]). RDEs are pervading in the mathematical modeling of the systems which
occur in biology, chemistry, complex physics phenomena, engineering, and
mechanics [4]. RDEs can also represent the chemical reactions and diffusion
processes. Basically many real-life processes do not only depend on the
present state but also past occurrences. Also, the dynamical systems are
constituted by the time delay. The study of nonlinear delay RDEs provides a
fundamental tool for quantitative and qualitative analyses of various
dynamical systems such as those modeling infections. For RDEs with delay, the
kinetic function $G$ which denotes the chemical reactions rates is a function
of both $u=u(x,t)$ and $w=u(x,t-\tau),$ which represent the sought after
concentration function and delayed argument, respectively. Two special cases
which can arise are $G(u,w)=g(w)$ and $G(u,w)=g(u).$ A system with local non-
equilibrium media is described by $G(u,w)=g(w).$ These are systems which
possess inertial properties and reactions will always begin after a time
$\tau.$ $G(u,w)=g(u)$ represents the classical local equilibrium case (See
e.g., [12]).

How to obtain the solutions of RDEs has recently attracted much attention.
NPDEs are universal in nature and for finding their solutions, several methods
have been employed which include spectral collocation and waveform relaxation
[13, 14, 15], adomian decomposition [16, 17], Tan-Cot [18], residual power
series [19], perturbation [20, 21]. However, there are disadvantages which are
commonly associated with those methods. There are conditions which make the
universal application of those listed methods and others to be impossible. The
objects are different in their geometric shapes. The reaction kinetics and
type of fluid flow are erratic. The worthlessness in the presence of singular
points is indisputable. Also, efforts have been made on using certain recently
proposed numerical and iterative methods, which are widely remarked for their
accuracy and effectiveness in obtaining the solutions of high-dimensional
complex geometry nonlinear problems (See e.g., [5, 6, 7, 8, 9, 10, 11]).
Obtaining the exact solutions is imperative for proper analysis of the
processes which are under consideration (localization, nonuniqueness, blowup
regimes, spatial, etc).

Subsequently, the term ”exact solution” in relating to NPDEs will refer to
where the solution can be expressed in:

  * (i)

terms of elementary functions;

  * (ii)

closed form with definite or/and indefinite integrals;

  * (iii)

terms of solutions to Ordinary Differential Equations (ODEs) or systems of
such equations.

Accepted form for exact solutions also includes the combinations of cases
listed above (See e.g, [12, 22, 23]). A most general form of (1.1) is when
nonlinear terms are present before the time derivative and the diffusion term.
An awkward form occurs when the combination of the kinetic function $G$ and
nonlinear terms are also associated with the variable derivative. This present
study is motivated by how to construct the exact solutions for a most general
form of nonlinear RDEs which are awkward in nature.

Let $G(u,w)$ be an arbitrary function which takes two arguments, $\tau>0$ be
the delay time, while $a(x),b(x),c(x),p(x)$ and $q(x)$ are some functions.
This study considers nonlinear delay RDEs of the form

$\begin{split}&\left[q(x)+c(x)g(u)\right]u_{t}=\left[a(x)u_{x}\right]_{x}\\\
&+\left[p(x)+b(x)G\left(u,w\right)\right]u_{x},~{}~{}w=u(x,t-\tau).\end{split}$
(1.2)

This is a most generalized form with the presence of variable coefficients and
arbitrary functions. The exact solutions are obtained in the form of
generalized traveling-wave solutions. A novel technique is presented for
constructing the exact solutions of a most generalized form of nonlinear RDEs
with variable coefficients and with delay. The exact solutions for coupled
reaction-diffusion nonlinear system with delay are also obtained. The
technique which is being presented in this paper can be applied for modeling a
wide class of problems, such as diffusion of pollutants and population models
due to the presence of arbitrary functions and free parameters. Besides it can
be very useful to generate the testing problems for numerical modeling.
The whole paper comprises of five sections. Section 1 contains a brief
introduction and literature survey. The rest of the manuscript is organized as
follows: How to construct the exact solutions in the form of a generalized
traveling-wave equation is presented in Section 2. Some examples are also
given in Section 2 to validate the study. Some generalization and required
transformations for the application of the proposed technique to more complex
partial differential equations are presented in Section 3. Some special cases
of and deductions from the main results of this study are discussed in Section
4. The study is substantiated in Section 5 with graphical analysis.

##  2 Solutions of generalized RDEs with delay

The aim is to find exact solutions of (1.2) which take the form

$u=U(y),~{}~{}~{}~{}y=t+\int h(x)dx,$ (2.1)

which is the generalized traveling-wave equation. Substitution of (2.1) into
(1.2) yields

$\begin{split}&a(x)h^{2}U^{\prime\prime}_{yy}+\left(\left[a(x)h\right]^{\prime}_{x}-q(x)+p(x)h\right)U^{\prime}_{y}\\\
&-c(x)g(U)U^{\prime}_{y}+b(x)hG\left(U,W\right)U^{\prime}_{y}=0,\end{split}$
(2.2)

where $W=U(y-\tau)$ and $h=h(x).$ The coefficients of the equation are chosen
such that they conform to the relations

$\displaystyle b(x)$ $\displaystyle=a(x)h(x),$ (2.3) $\displaystyle c(x)$
$\displaystyle=-a(x)h^{2}(x),$ (2.4) $\displaystyle q(x)$
$\displaystyle=\left[a(x)h(x)\right]^{\prime}_{x}+p(x)h(x)+ka(x)h^{2}(x),$
(2.5)

where $k$ is a constant. The relations reduce (2.2) to

$U^{\prime\prime}_{yy}+\left[G\left(U,W\right)+g(U)-k\right]U^{\prime}_{y}=0,~{}~{}~{}W=U(y-\tau),$
(2.6)

which is a delay ODE. A standard form for (2.5) is

$a(x)h^{\prime}_{x}+ka(x)h^{2}+\left[p(x)+a^{\prime}(x)\right]h-q(x)=0.$ (2.7)

Given the functions $a(x),p(x)$ and $q(x),$ the relation (2.7) is the Riccati
ODE for $h=h(x).$

###### Remark 2.1

Solving the Riccati equation (2.7) is not necessary for a given $h=h(x).$ Then
the derived generalized traveling-wave equation (2.1), is said to be the exact
solutions to certain RDEs with delay of the form (1.2) for which (2.3), (2.4)
and (2.5) are true.

Consideration is given to the Riccati ODE (2.7) under two cases, which are
degenerate and nondegenerate.
Degenerate case. For $k=0,$ the degeneration of Riccati equation (2.7) has the
general solution which is given by

$h(x)=\frac{\int\nu(x)q(x)dx+C_{1}}{a(x)\nu(x)},~{}~{}~{}\nu(x)=Exp\left(\int\frac{p(x)}{a(x)}dx\right),$
(2.8)

where $C_{1}$ denotes an arbitrary constant.

###### Example 2.1

Consider the case where $a=1$ and $p(x)=q(x)=x.$ Using (2.8) with $C_{1}=0,$
gives $h(x)=1.$ Substitution of the value of $h$ into (2.1), (2.3) and (2.4)
yields $y=t+x,b=1$ and $c=-1.$ Consequently, for functions $g(u)$ and $G(u,w)$
which are arbitrary, the nonlinear RDE with delay

$\left[x-g(u)\right]u_{t}=u_{xx}+\left[x+G\left(u,w\right)\right]u_{x},~{}~{}w=u(x,t-\tau),$

is solved by

$u=U(y),~{}~{}~{}~{}y=t+x,$

where $U(y)$ is determined by the delay ODE

$U^{\prime\prime}_{yy}+\left[G\left(U,W\right)+g(U)\right]U^{\prime}_{y}=0,~{}~{}~{}W=U(y-\tau),$
(2.9)

which is obtained from (2.6) by setting $k=0.$

###### Example 2.2

Consider the case where $a=q=1$ and $p(x)=\frac{1}{x}.$ Using (2.8) with
$C_{1}=0,$ gives $h(x)=\frac{x}{2}.$ Substitution of the value of $h$ into
(2.1), (2.3) and (2.4) yields $y=t+\frac{x^{2}}{4},b=\frac{x}{2}$ and
$c=-\frac{x^{2}}{4}.$ Consequently, for functions $g(u)$ and $G(u,w)$ which
are arbitrary, the nonlinear RDE with delay

$\left[4-x^{2}g(u)\right]u_{t}=4u_{xx}+\left[\frac{4}{x}+2xG\left(u,w\right)\right]u_{x},~{}~{}w=u(x,t-\tau),$

is solved by

$u=U(y),~{}~{}~{}~{}y=t+\frac{x^{2}}{4},$

where $U(y)$ is determined by (2.9).

###### Example 2.3

Consider the case where $a=1,p=0,$ and $q(x)=x.$ Using (2.8) with $C_{1}=0,$
gives $h(x)=\frac{1}{2}x^{2}.$ Substitution of the value of $h$ into (2.1),
(2.3) and (2.4) yields $y=t+\frac{1}{6}x^{3},b=\frac{1}{2}x^{2}$ and
$c=-\frac{1}{4}x^{4}.$ Consequently, for functions $g(u)$ and $G(u,w)$ which
are arbitrary, the nonlinear RDE with delay

$\left[4x-x^{4}g(u)\right]u_{t}=4u_{xx}+2x^{2}G\left(u,w\right)u_{x},~{}~{}w=u(x,t-\tau),$

is solved by

$u=U(y),~{}~{}~{}~{}y=t+\frac{1}{6}x^{3},$

where $U(y)$ is determined by (2.9).

###### Example 2.4

Consider the case where $p=0,$ and $a(x)=q(x)=x.$ Using (2.8) with $C_{1}=0,$
gives $h(x)=\frac{1}{2}x.$ Substitution of the value of $h$ into (2.1), (2.3)
and (2.4) yields $y=t+\frac{1}{4}x^{2},b=\frac{1}{2}x$ and
$c=-\frac{1}{4}x^{2}.$ Consequently, for functions $g(u)$ and $G(u,w)$ which
are arbitrary, the nonlinear RDE with delay

$\left[x-\frac{1}{4}x^{2}g(u)\right]u_{t}=\left[xu_{x}\right]_{x}+\frac{1}{2}xG\left(u,w\right)u_{x},~{}~{}w=u(x,t-\tau),$

is solved by

$u=U(y),~{}~{}~{}~{}y=t+\frac{1}{4}x^{2},$

where $U(y)$ is determined by (2.9).

###### Example 2.5

Consider the case where two constant coefficients are given as $p(x)=1$ and
$q(x)=0,$ while the third coefficient is given arbitrarily as $a=a(x).$ Using
(2.8) with $C_{1}=0,$ gives $h(x)=\frac{x}{a(x)}.$ Substitution of the value
of $h$ into (2.1), (2.3) and (2.4) yields $y=t+\int\frac{x}{a(x)}dx,b=x$ and
$c=-\frac{x^{2}}{a(x)}.$ Consequently, for functions $g(u)$ and $G(u,w)$ which
are arbitrary, the nonlinear RDE with delay

$\frac{x^{2}}{a(x)}g(u)u_{t}+\left[a(x)u_{x}\right]_{x}+\left[1+xG\left(u,w\right)\right]u_{x}=0,$
(2.10)

$w=u(x,t-\tau),$ is solved by

$u=U(y),~{}~{}~{}~{}y=t+\int\frac{x}{a(x)}dx,$

where $U(y)$ is determined by (2.9). Substitution of the function $a(x)=x^{n}$
into (2.10) produces a nonlinear RDE with delay

$x^{2-n}g(u)u_{t}+\left[x^{n}u_{x}\right]_{x}+\left[1+xG\left(u,w\right)\right]u_{x}=0,~{}~{}w=u(x,t-\tau),$

where $n$ represents any number.

Nondegenerate case. When $k\neq 0,$ let

$h=\frac{1}{k}\frac{{\psi}^{\prime}_{x}}{{\psi}}.$ (2.11)

Substituting (2.11) into (2.7) produces

$\frac{a(x)}{k}\left(\frac{{\psi}^{\prime\prime}_{xx}}{{\psi}}-\frac{{\psi}^{\prime}_{x}}{{\psi}^{2}}\right)+ka(x)\left(\frac{1}{k}\frac{{\psi}^{\prime}_{x}}{{\psi}}\right)^{2}+\left[p(x)+a^{\prime}(x)\right]\frac{1}{k}\frac{{\psi}^{\prime}_{x}}{{\psi}}-q(x)=0,$

which is simplified to obtain linear ODE of second-order

$a(x){\psi}^{\prime\prime}_{xx}+\left[p(x)+a^{\prime}(x)\right]{\psi}^{\prime}_{x}-kq(x){\psi}=0.$
(2.12)

Interested readers in the exact solutions of (2.12) for varieties of the
functions $a(x),p(x)$ and $q(x),$ are referred to [24, 25].

###### Example 2.6

Consider the case where $a=q=1$ and $p=0.$ The equation (2.12) has the general
solution which is given by

${\psi}=\begin{cases}{\Lambda}_{1}~{}cos(\lambda
x)+{\Lambda}_{2}~{}sin(\lambda x),&{\rm}\
\phantom{\infty}\text{if}\,\,k=-{\lambda}^{2}<0,\\\
{\Lambda}_{1}~{}cosh(\lambda x)+{\Lambda}_{2}~{}sinh(\lambda x),&{\rm}\
\phantom{\infty}\text{if}\,\,k={\lambda}^{2}>0,\\\ \end{cases}$ (2.13)

where ${\Lambda}_{1}$ and ${\Lambda}_{2}$ are arbitrary constants. Setting
${\Lambda}_{1}=1,{\Lambda}_{2}=0,$ and $k=-1~{}\left(<0\right)$ in (2.13), and
using formula (2.11) gives

$h(x)=tan~{}x.$

Substituting the function $h$ into (2.3) and (2.4) gives

$b(x)=tan~{}x,c(x)=-{tan}^{2}x.$

Hence, for arbitrary functions $g(u)$ and $G(u,w),$ the nonlinear RDE with
delay

$\left[1-{tan}^{2}x~{}g(u)\right]u_{t}=u_{xx}+tan~{}x~{}G\left(u,w\right)u_{x},$
(2.14)

is solved by

$u=U(y),y=t-In~{}cos~{}x,$ (2.15)

where $U(y)$ is determined by the delay ODE

$U^{\prime\prime}_{yy}+\left[G\left(U,W\right)+g(U)+1\right]U^{\prime}_{y}=0,~{}~{}~{}W=U(z-\tau).$
(2.16)

## Constructing exact solutions when the function $h(x)$ is given

Table 1 gives the list of other possible ways which do not involve integrating
the Riccati equation (2.7), for constructing solutions which are exact to
equations of the form (1.2). Among the functions $h(x),a(x),p(x),$ and $q(x),$
three functions which include $h(x)$ are assumed to be given. The unknown
function which remains is then derived from (2.7). The whole nonlinear RDE of
the form (1.2) is determined by using (2.3) and (2.4).

Table 1: Possible ways of deriving the unknown function when out of the functions $h(x),a(x),p(x),$ and $q(x),$ three functions which include $h(x)$ are given. Note that $k$ and $C_{2}$ are arbitrary constants. No. |  Given functions |  Derived function
---|---|---
1 |  $h=h(x),a=a(x),$ |  $q(x)=\left[ah\right]^{\prime}_{x}+ph+kah^{2}.$
$p=p(x).$
2 |  $h=h(x),a=a(x),$ |  $p(x)=\left(q-ah^{\prime}_{x}-kah^{2}\right)/h-a^{\prime}.$
$q=q(x).$
3 |  $h=h(x),p=p(x),$ |  $a(x)=\frac{\int\mu\left(q/h-p\right)dx+C_{2}}{\mu},$
$q=q(x).$ |  where $\mu=h\ \mbox{exp}(k\int hdx).$

###### Example 2.7

Illustration is given by using the third way of deriving the unknown function
which is described in Table 1. The $h=h(x)$ is arbitrary, while $C_{2}=0,q=0,$
and $p=-1.$ Two possible cases are being considered.

  * (I)

Degenerate case $k=0.$ It is obtained that $a(x)=\int hdx/h.$ Consequently,
for functions $g(u)$ and $G(u,w)$ which are arbitrary, the nonlinear RDE with
delay

$c(x)g(u)u_{t}=\left[a(x)u_{x}\right]_{x}+\left[b(x)G\left(u,w\right)-1\right]u_{x},$
(2.17)

where $a(x)=\int hdx/h,b(x)=\int hdx,$ and $c(x)=-h\int hdx,$ is solved by

$u=U(y),~{}y=t+\int hdx.$ (2.18)

Here, $U(y)$ in (2.18) is determined by (2.9).

  * (II)

Nondegenerate case $k\neq 0.$ A new function $f=f(x)$ is introduced by setting
$f(x)=\mbox{exp}(k\int hdx).$ Therefore, it is obtained that $a(x)=\int
hfdx/hf.$ Consequently, for arbitrary functions $g(u)$ and $G(u,w),$ the
equation (2.17), where $a(x)=\int hfdx/hf,b(x)=\int hfdx/f,$ and $c(x)=-h\int
hfdx/f,$ is solved by (2.18). Here, $U(y)$ is determined by (2.6).

###### Example 2.8

An elucidation is given for the third way of deriving the unknown function
which is described in Table 1. Consider Example 2.7 with $h(x)=1/x.$

Degenerate case ($k=0$): For functions $g(u)$ and $G(u,w)$ which are
arbitrary, the nonlinear RDE

$ln\left(1/\sqrt[x]{x}\right)g(u)u_{t}=\left[x~{}lnx~{}u_{x}\right]_{x}+\left[lnx~{}G\left(u,w\right)-1\right]u_{x},$

is solved by

$u=U(y),~{}y=t+ln~{}x,$

where $U(y)$ is determined by (2.9).

Nondegenerate case ($k\neq 0$): For arbitrary functions $g(u)$ and $G(u,w),$
the equation

$\frac{1}{kx}g(u)u_{t}+\left[\frac{x}{k}u_{x}\right]_{x}+\left[\frac{1}{k}G\left(u,w\right)-1\right]u_{x}=0,$

is solved by (2.18). Here, $U(y)$ is determined by (2.6).

##  3 Some transformations and generalization

The following parameters are assumed to be chosen such that the below
relations are satisfied:

$\displaystyle b(x,t)$ $\displaystyle=$ $\displaystyle a(x,t)h(x),$
$\displaystyle c(x,t)$ $\displaystyle=$ $\displaystyle-a(x,t)h^{2}(x),$ (3.1)
$\displaystyle q(x,t)$ $\displaystyle=$
$\displaystyle\left[a(x,t)h(x)\right]^{\prime}_{x}+p(x,t)h(x)$
$\displaystyle+ka(x,t)h^{2}(x),$

where the constant is $k.$

  * (I)

One spatial dimension nonlinear RDEs type with delay

Consider the nonlinear RDEs with delay which is given as

$\begin{split}&\left[q(x,t)+c(x,t)g(u)\right]u_{t}=\left[a(x,t)u_{x}\right]_{x}\\\
&+p(x,t)u_{x}+b(x,t)h(x)G\left(h,u,w,u_{x}/h\right),\end{split}$ (3.2)

where the arbitrary function $G\left(h,u,w,\theta\right)$ takes four
arguments, $w=u(x,t-\tau)$ and the delay in time is $\tau>0.$ The equation
(3.2) has

$u=U(y),~{}y=t+\int h(x)dx,$ (3.3)

as its general solution. Here, $U(y)$ is determined by the delay ODE

$U^{\prime\prime}_{yy}+(g(U)-k)U^{\prime}_{y}+G\left(h,U,W,U^{\prime}_{y}\right)=0,~{}~{}~{}W=U(z-\tau).$

This can be easily verified by substituting the functions in (3.3) into (3.2),
while taking into account the relations (3).

###### Example 3.1

An illustration is given for the degenerate case $k=0.$ In the third relation
of equation (3), set $a(x,t)=a(x),p(x,t)=0,$ and $q(x,t)=1.$ It is obtained
that $b(x,t)=x,c(x,t)=-\frac{x^{2}}{a(x)}$ and $h(x)=\frac{x}{a(x)}.$ Also in
(3.2), setting
$G\left(h,u,w,\theta\right)=\varphi(u){\theta}+h\psi(u,w){\theta}^{2}$ leads
to the deduction that the nonlinear RDE with delay which is of the form

$\left[1-\frac{x^{2}}{a(x)}g(u)\right]u_{t}=\left[a(x,t)u_{x}\right]_{x}+x\varphi(u)u_{x}+x\psi(u,w)u^{2}_{x},$

is solved by

$u=U(y),~{}z=t+\int\frac{x}{a(x)}dx,$

where $U(y)$ is determined by the delay ODE

$U^{\prime\prime}_{yy}+g(U)U^{\prime}_{y}+G\left(h,U,W,U^{\prime}_{y}\right)=0.$

###### Remark 3.1

For certain functions $a(x,t),p(x,t),$ and $q(x,t),$ exact solutions of the
degenerate case ($k=0$) for the third equation in the relation (3)can be found
in [26].

  * (II)

Systems of nonlinear RDEs with delay

Consider the system of nonlinear RDEs with delay which is given as

$\begin{split}&\left[q(x,t)+c(x,t)g_{1}(u,v)\right]u_{t}=\left[a(x,t)u_{x}\right]_{x}\\\
&+p(x,t)u_{x}+b(x,t)h(x)G_{1}\left(h,u,v,w_{1},u_{x}/h,v_{x}/h\right),\\\ &\\\
&\left[q(x,t)+c(x,t)g_{2}(u,v)\right]v_{t}=\left[a(x,t)v_{x}\right]_{x}\\\
&+p(x,t)v_{x}+b(x,t)h(x)G_{2}\left(h,u,v,w_{2},u_{x}/h,v_{x}/h\right),\end{split}$
(3.4)

where $G_{1}\left(h,u,v,w_{1},{\theta}_{1},{\theta}_{2}\right),$
$G_{2}\left(h,u,v,w_{2},{\theta}_{1},{\theta}_{2}\right)$ are arbitrary
functions which take six arguments, $w_{1}=u(x,t-\tau),~{}w_{2}=v(x,t-\tau),$
and the delay in time is $\tau>0.$ The system of equations (3.4) is solved by

$u=U(y),v=V(y),~{}y=t+\int h(x)dx.$ (3.5)

Here, $U(y)$ and $V(y)$ are determined by the coupled delay ODE

$\displaystyle
U^{\prime\prime}_{yy}+(g_{1}(U,V)-k)U^{\prime}_{y}+G\left(h,U,V,W_{1},U^{\prime}_{y},V^{\prime}_{y}\right)$
$\displaystyle=0,$ $\displaystyle
V^{\prime\prime}_{yy}+(g_{2}(U,V)-k)V^{\prime}_{y}+G\left(h,U,V,W_{2},U^{\prime}_{y},V^{\prime}_{y}\right)$
$\displaystyle=0,$

where $W_{1}=U(y-\tau)$ and $W_{2}=V(y-\tau).$ This can be easily verified by
substituting the given functions in (3.5) into (3.4), while taking into
account the relations (3).

###### Example 3.2

An example is given for the highlight of the degenerate case $k=0.$ This is
done by setting $a(x,t)=1,~{}q(x,t)=2,$ and $p(x,t)=0$ in the third relation
of equation (3). It is obtained that $h(x)=2x,b(x,t)=2x$ and $c(x,t)=-4x^{2}.$
By setting
$G_{1}\left(h,u,v,w_{1},{\theta}_{1},{\theta}_{2}\right)=hf_{1}(u,v,w){\theta}_{1}^{2},$
$G_{2}\left(h,u,v,w_{1},{\theta}_{1},{\theta}_{2}\right)=hf_{2}(u,v,w){\theta}_{2}$
in (3.4), the coupled nonlinear reaction-diffusion system with delay

$\displaystyle\left[2-4x^{2}g(u,v)\right]u_{t}$
$\displaystyle=u_{xx}+2xf_{1}(u,v,w)u^{2}_{x},$
$\displaystyle\left[2-4x^{2}g(u,v)\right]v_{t}$
$\displaystyle=v_{xx}+4x^{2}f_{2}(u,v,w)v_{x},$

is solved by

$u=U(y),v=V(y),~{}y=t+x^{2},$

where $U(y)$ and $V(y)$ are determined by the coupled ODEs with delay

$\displaystyle
U^{\prime\prime}_{yy}+g_{1}(U,V)U^{\prime}_{y}+G_{1}\left(h,U,V,W,U^{\prime}_{y},V^{\prime}_{y}\right)$
$\displaystyle=0,$ $\displaystyle
V^{\prime\prime}_{yy}+g_{1}(U,V)U^{\prime}_{y}+G_{2}\left(h,U,V,W,U^{\prime}_{y},V^{\prime}_{y}\right)$
$\displaystyle=0.$

##  4 Some transformations and deductions

Special cases of (1.2) are being discussed in this section.

  * (I)

Consider the case $c(x)=p(x)=0.$

The nonlinear delay RDE

$q(x)u_{t}=\left[a(x)u_{x}\right]_{x}+b(x)G\left(u,w\right)u_{x},~{}~{}w=u(x,t-\tau),$
(4.1)

is obtained. Some existing results can be obtained as corollaries to (4.1)
(See e.g, [22]).

  * (II)

Consider the case $c(x)=0$ and $G\left(u,w\right)=g(u).$

Nonlinear RDE of the form

$q(x)u_{t}=\left[a(x)u_{x}\right]_{x}+p(x)u_{x}+b(x)g(u)$ (4.2)

is obtained. The equation (4.2) is a corollary to our results and it
represents the classical local equilibrium case. This vindicates that our
results are extension of some existing results where such cases were
considered (See e.g, [23]).

##  5 Graphical illustration

Examples are given to show the graphical illustration of the solutions of
certain RDEs.

###### Example 5.1

Consider the RDE

$x^{2}u_{t}=u_{xx}-xuu_{x}.$ (5.1)

The interval is $0\leq x\leq 1,$ where the time $t\geq 0.$ The initial
condition which is the solution at $t=0$ is given as

$u(x,0)=sin(\pi x).$

Moreover, the Dirichlet boundary conditions are given as

$u(0,t)=0\ \mbox{and}\ \ u(1,t)=0.$

Figure 1 shows the graphical illustration for (5.1). It displays how $u$
changes with respect to $x$ at $t=2.$

Figure 1: Solutions of $x^{2}u_{t}=u_{xx}-xuu_{x}.$

###### Example 5.2

The convection-diffusion equation is given as

$u_{t}=u_{xx}+b(x)u_{x},$ (5.2)

which has the initial condition as $u(0,x)=\frac{1}{1+(x-5)^{2}}$ and the
Dirichlet boundary conditions $u(t,-\infty)=u(t,+\infty)=0,$ where

$b(x)=3\bar{u}(x)^{2}-2\bar{u}(x)$

and $\bar{u}$ is implicitly defined by the relation

$\frac{1}{\bar{u}}+log\frac{1-\bar{u}}{\bar{u}}=x.$

The function $\bar{u}$ is known as an equilibrium solution of

$u_{t}+\left(u^{3}-u^{2}\right)_{x}=u_{x}x,$

where $\bar{u}(-\infty)=1$ and $\bar{u}(+\infty)=0$ (See e.g [27]). Figure 2
shows the graphical illustration for (5.2). A peak is observed to occur at
$x=5$ as changes in $u$ with respect to $x$ is being displyed at $t=2.$

Figure 2: Solutions of $u_{t}=u_{xx}+b(x)u_{x}$ (5.2).

###### Example 5.3

Consider a system of RDEs

$\begin{split}u_{t}&=u_{xx}+(1-u-v)u_{x},\\\ v_{t}&=v_{xx}+(1-u-v)v_{x},\\\
\end{split}$ (5.3)

with the initial conditions

$\begin{cases}u(0,x)=x^{2},\\\ v(0,x)=x(x-2),\\\ \end{cases}$

and Cauchy boundary conditions which are specified as

$\begin{cases}u_{x}(t,0)=0;~{}u(t,1)=1,\\\ v(t,0)=0;~{}v_{x}(t,1)=0.\\\
\end{cases}$

Figure 3: Solutions for $u$ in the system of reaction-diffusion (5.3). Figure
4: Solutions for $v$ in the system of reaction-diffusion (5.3).

For the system of reaction-diffusion (5.3), the solution for $u$ is displayed
in Figure 3 while the solution for $v$ is displayed in Figure 4. Observe that
while $u$ has a positive slope, the slope of $v$ is negative.

##  6  Conclusion

Recently, exact solutions of RDEs and reaction-diffusion systems have
attracted great attention. In this paper, exact solutions are presented for a
most generalized form of RDE with delay and which are associated with variable
coefficients. The presence of arbitrary functions and free parameters in the
solutions represents their feasible application in solving certain modeling
problems such as diffusion of pollutants and population models, where the
population is spatially distributed. It also makes the obtained solutions to
be suitable for usage in testing the numerical and approximate analytical
methods. The obtained results also find applications in finding the exact
solutions of other form of partial differential equations which are more
complex. The examples of delay RDEs with their exact solutions are displayed
for elucidation. Graphical illustration of the solutions of some specific RDEs
are given.

Abbreviations
NPDEs: Nonlinear Partial Differential Equations
RDEs: Reaction-Diffusion Equations

## 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.

## Acknowledgements

The first author acknowledges with thanks the postdoctoral fellowship and
financial support from the DSI-NRF Center of Excellence in Mathematical and
Statistical Sciences (CoE-MaSS). Opinions expressed and conclusions arrived
are those of the authors and are not necessarily to be attributed to the CoE-
MaSS.

##  References

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https://www.math.tamu.edu/ phoward/m442/pdemat.pdf


# Global existence and multiplicity of solutions for logarithmic Schrödinger
equations on graphs

Mengqiu Shao<EMAIL_ADDRESS>School of Mathematical Sciences, Qufu Normal
University, Shandong, 273165, China

###### Abstract

We consider the following logarithmic Schrödinger equation

$-\Delta u+h(x)u=u\log u^{2}$

on a locally finite graph $G=(V,E)$, where $\Delta$ is a discrete Laplacian
operator on the graph, $h$ is the potential function. Different from the
classical methods in Euclidean space, we obtain the existence of global
solutions to the equation by using the variational method from local to
global, which is inspired by the works of Lin and Yang in [17]. In addition,
when the potential function $h$ is sign-changing, we prove that the equation
admits infinitely many solutions with high energy by using the symmetric
mountain pass theorem. We extend the classical results in Euclidean space to
discrete graphs.

###### keywords:

logarithmic Schrödinger equation; global solutions; multiple solutions

###### MSC:

[2020] 35A15, 35Q55, 58E30

††journal: ***

##  1 Introduction

In recent years, the partial differential equation on graphs has become a hot
topic in the field of discrete geometry and nonlinear analysis. For example,
Lin and Wu [16] studied the existence and nonexistence of global solutions for
a semilinear heat equation on graphs. Hua, Keller, Schwarz and Wirth [12]
investigated the eigenvalue growth of infinite graphs with discrete spectrum.
Cushing, Kamtue, Liu and Peyerimhoff [5] reformulated the Bakry-Émery
curvature on a weighted graph. Moreover, the existence and asymptotic behavior
of solutions to elliptic equations have been widely studied on graphs, see
[14, 15] for more details. In particular, Lin and Yang [17] considered various
equations on locally finite graphs and established a variational method from
local to global on graphs. More recently, Chang, Wang and Yan [7] first
investigated the existence of ground state solutions of a Schrödinger equation
with logarithmic nonlinearity on locally finite graphs. Later, Chang et al.
[8] considered the existence of least energy sign-changing solutions of a
logarithmic Schrödinger equation on locally finite graphs. The above works
arouse our great research interest. In this paper, we are focus on the
following logarithmic Schrödinger equation

$\displaystyle-\Delta u+h(x)u=u\log u^{2}$ (1)

on a locally finite graph $G=(V,E)$, where $\Delta$ is the discrete Laplacian
on graphs and $h$ is the potential function satisfying certain conditions. On
the Euclidean space, equation (1) is closely related to the time-dependent
logarithmic Schrödinger equation

$i\frac{\partial u}{\partial t}-\Delta u+h(x)u-u\log u^{2}=0\ \ \hbox{in}\ \
\mathbb{R}^{N}\times\mathbb{R}_{+},$

which has important applications in quantum mechanics, quantum optics, nuclear
physics, transport and diffusion phenomena, open quantum systems, effective
quantum gravity, theory of superfluidity, and Bose-Einstein condensation (see
[4, 6, 23, 25] for more details).

When $V=\mathbb{R}^{N}$, the equation (1) is formally associated with the
energy functional
$J:W^{1,2}(\mathbb{R}^{N})\rightarrow\mathbb{R}\cup\\{+\infty\\}$ defined by

$J(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx+\frac{1}{2}\int_{\mathbb{R}^{N}}(h(x)+1)u^{2}dx-\frac{1}{2}\int_{\mathbb{R}^{N}}u^{2}\log
u^{2}dx.$

However, in general, this functional fails to be finite and $C^{1}$ smooth on
$W^{1,2}(\mathbb{R}^{N})$ because there exists $u\in W^{1,2}(\mathbb{R}^{N})$
such that $\int_{\mathbb{R}^{N}}u^{2}\log u^{2}dx=-\infty$. Thus the classical
critical point theory cannot be applied for $J$. To overcome this obstacle,
several approaches have been developed on the Euclidean space. For instance,
Cazenave [6] worked in an Orlicz space endowed with Luxemburg type norm in
order to make the associated functional well defined and $C^{1}$ smooth. By
applying non-smooth critical point theory for lower semi-continuous
functionals, Squassina and Szulkin [20] proved the existence of multiple
solutions to a logarithmic Schrödinger equation with periodic potential. By
using the penalization technique, Tanaka and Zhang [21] obtained infinitely
many multi-bump geometrically distinct solutions for a logarithmic Schrödinger
equation. For a more non-smooth variational framework to the logarithmic
Schrödinger equation, we refer the reader to[1, 2, 3, 9, 11, 19].

It is worth pointing out that the general form of the equation (1) is the
following Schrödinger equation

$-\Delta u+h(x)u=f(x,u),\ \ x\in V.$ (2)

On a lattice graphs, when the potential function $h$ is periodic or bounded,
Hua and Xu [13] proved the existence of ground state solutions for the
equation (2) by using the Nehari manifold method. When $h(x)=\lambda a(x)+1$
and $f(x,u)=|u|^{p-1}u$, $p\geq 2$, Zhang and Zhao [24] proved the existence
and convergence of ground state solutions to the equation (2) on locally
finite graphs. Under the following Ambrosetti-Rabinowitz (or (A-R) for short)
condition $(f_{1})$ _there exists a constant $\alpha>2$ such that for all
$x\in V$ and $s\in\mathbb{R}\setminus\\{0\\}$,_

$0<\alpha F(x,s)=\alpha\int^{s}_{0}f(x,t)dt\leq sf(x,s)$

and some growth conditions, Grigor’yan, Lin and Yang [10] obtained that the
equation (2) admits a positive solution on graphs, where the (A-R) condition
$(f_{1})$ plays an important role in proving the boundedness of $(PS)$
sequence.

However, the logarithmic nonlinearity $u\log u^{2}$ in the equation (1) not
satisfies the (A-R) condition, which brings some difficulties when we use the
variational method to study such equations on graphs. On the other hand,
similarly to the case in Euclidean space, the corresponding energy functional
of the equation (1) is not well defined on $W^{1,2}(V)$. Particularly, Chang,
Wang and Yang [7] gave the following example

$u(x)=\left\\{\begin{array}[]{lll}(|x|\log|x|)^{-1},&|x|\geq 3,\\\\[6.45831pt]
0,&|x|\leq 2\end{array}\right.$

and verified $u\in W^{1,2}(V)$ but $\int_{V}u^{2}\log u^{2}d\mu=-\infty$,
where $x_{0}\in V$ is some fixed point and $|x|:=\rho(x,x_{0})$ denotes the
distance between $x$ and $x_{0}$ on graphs. To deal with this limitation, they
restricted $u^{2}\log u^{2}\in L^{1}(V)$ and applied the direction derivative
and the Nehari manifold method to obtain the existence of ground state
solutions of the equation (1) in [7].

Inspired by the above works, this paper is devoted to study the existence and
multiplicity of solutions to the logarithmic Schrödinger equation (1) on the
locally finite graph $G=(V,E)$. Specifically, our first aim is to obtain the
existence of global solutions to the equation (1) by using the variational
method from local to global, which is different from the classical methods in
Euclidean space and the analysis techniques in [7]. Roughly speaking, we first
get sequences of local solutions $u_{k}$ for equation (1) on $W^{1,2}(B_{k})$
and derive uniform estimates for those local solution sequences. Then we
obtain the global solutions by extracting convergent sequence of local
solutions. We have to point out that one of the difficulties is to prove the
nontrivial property of the global solutions. On the other hand, under the
assumption that the potential function $h$ allows changing sign, we also prove
that the equation (1) admits infinitely many solutions with high energy by
using the symmetric mountain pass theorem. To the best of our knowledge, there
seem no results concerned with infinitely many solutions to such equations on
graphs.

To describe our main results in details, we first give some notations and
assumptions. Throughout this paper, we always assume that $G=(V,E)$ is a
connected locally finite graph and its measure
$\mu:V\rightarrow\mathbb{R}^{+}$ is a finite positive function satisfying

$\mu(x)\geq\mu_{min}>0\ \ \hbox{for\ \ all}\ \ x\in V.$ (3)

For an edge $xy\in E$, the weight $\omega_{xy}$ is positive and symmetric,
namely

$\omega_{xy}=\omega_{yx}>0.$ (4)

The distance $\rho(x,y)$ of two vertices $x,y\in V$ is defined by the minimal
number of edges which connect these two vertices. For any function
$u:V\rightarrow\mathbb{R}$ and $x\in V$, the Laplacian of $u$ at $x$ is
defined by

$\Delta u(x):=\frac{1}{\mu(x)}\underset{y\sim x}{\sum}\omega_{xy}(u(y)-u(x)).$

For any fixed $x\in V$, the gradient of $u$ at $x$ is given by

$\nabla
u(x)=\left(\sqrt{\frac{\omega_{xy_{1}}}{2\mu(x)}}(u(y_{1})-u(x)),\cdots,\sqrt{\frac{\omega_{xy_{\ell_{x}}}}{2\mu(x)}}(u(y_{\ell_{x}})-u(x))\right),$

where $\\{y_{1},\cdots,y_{\ell_{x}}\\}$ is the set of all neighbors of $x$ for
some positive integer $\ell_{x}$. Then $\nabla u(x)$ can be seen as a vector
in $\mathbb{R}^{\ell_{x}}$. $\nabla u\nabla v$ at $x\in V$ is denoted by

$\nabla u\nabla v(x)=\frac{1}{2\mu(x)}\underset{y\sim
x}{\sum}\omega_{xy}(u(y)-u(x))(v(y)-v(x)).$

The length of the gradient for $u$ is denoted by

$|\nabla u(x)|=\left(\frac{1}{2\mu(x)}\underset{y\sim
x}{\sum}\omega_{xy}(u(y)-u(x))^{2}\right)^{\frac{1}{2}}.$

For any function $u:V\rightarrow\mathbb{R}$, an integral of $u$ over $V$ is
defined by

$\int_{V}ud\mu=\underset{x\in V}{\sum}\mu(x)u(x).$

For any $1\leq s<\infty$, we denote by

$L^{s}(V):=\left\\{u:V\rightarrow\mathbb{R}:\int_{V}|u|^{s}d\mu<\infty\right\\}$

the set of integrable functions on $V$ with the respect to the measure $\mu$.
For $s=\infty$, let

$L^{\infty}(V):=\left\\{u:V\rightarrow\mathbb{R}:\underset{x\in
V}{\sup}|u(x)|<\infty\right\\}.$

Usually, we use $\|\cdot\|_{s}$ to denote the $L^{s}(V)$ norm. Define

$W^{1,2}(V):=\left\\{u:V\rightarrow\mathbb{R}:\int_{V}(|\nabla
u|^{2}+u^{2})d\mu<+\infty\right\\}$

with the norm

$\|u\|_{W^{1,2}(V)}:=\left(\int_{V}(|\nabla
u|^{2}+u^{2})d\mu\right)^{\frac{1}{2}}.$

Clearly, $W^{1,2}(V)$ is a Hilbert space with the inner product

$(u,v)_{W^{1,2}(V)}=\int_{V}(\nabla u\nabla v+uv)d\mu,\ \ \forall u,v\in
W^{1,2}(V).$

Let $h(x)\geq h_{0}>0$ for all $x\in V$, we define a subspace of $W^{1,2}(V)$,
which is also a Hilbert space, namely

$\mathcal{H}:=\\{u\in W^{1,2}(V):\int_{V}h(x)u^{2}d\mu<+\infty\\}$ (5)

with the norm

$\|u\|_{\mathcal{H}}:=\left(\int_{V}(|\nabla
u|^{2}+h(x)u^{2})d\mu\right)^{\frac{1}{2}}.$

Our first result is as follows.

######  Theorem 1.1.

Let $G=(V,E)$ be a connected locally finite graph satisfying (3) and (4).
Assume that $h$ satisfies the following conditions:

  1. ($h_{1}$)

there exists a constant $h_{0}>0$ such that $h(x)\geq h_{0}$ for all $x\in V$;

  2. ($h_{2}$)

$h^{-1}\in L^{1}(V)$.

Then the equation (1) possesses a nontrivial solution.

######  Remark 1.2.

_The assumptions $(h_{1})$ and $(h_{2})$ were originally introduced on graphs
by Grigor’yan, Lin and Yang in [10] which is used to guarantee the compactness
of the embedding $\mathcal{H}\hookrightarrow L^{s}(V)$, $s\in[1,+\infty]$.
Especially, this embedding theorem plays an important role in proving the
nontrivial property of the global solutions._

Moreover, we consider the existence of infinitely many solutions to the
equation (1) when $h$ satisfies the following assumptions.

  1. ($h^{\prime}_{1}$)

_$\underset{x\in V}{\inf}h(x)\geq h_{1}$ for some constant $h_{1}\in(-1,0)$._

  2. ($h^{\prime}_{2}$)

_There exists $\alpha>0$ such that $h^{-1}\in L^{1}(V\setminus V_{\alpha})$,
where $V_{\alpha}=\\{x\in V:h(x)\leq\alpha\\}$ and the volume of $V_{\alpha}$
is finite, i.e._

$Vol(V_{\alpha})=\underset{x\in V_{\alpha}}{\sum}\mu(x)<\infty.$

Note that the equation (1) is formally associated with the energy functional
$\mathcal{J}:W^{1,2}(V)\rightarrow\mathbb{R}\cup\\{+\infty\\}$ defined by

$\mathcal{J}(u)=\frac{1}{2}\int_{V}|\nabla
u|^{2}d\mu+\frac{1}{2}\int_{V}(h(x)+1)u^{2}d\mu-\frac{1}{2}\int_{V}u^{2}\log
u^{2}d\mu.$ (6)

To study the multiplicity of solutions to the equation (1), we consider a
function space

$\mathcal{W}:=\\{u\in W^{1,2}(V):\int_{V}(h(x)+1)u^{2}d\mu<+\infty\\}$ (7)

with the norm

$\|u\|_{\mathcal{W}}:=\left(\int_{V}[|\nabla
u|^{2}+(h(x)+1)u^{2}]d\mu\right)^{\frac{1}{2}},$

where $h$ satisfies $(h^{\prime}_{1})$ and $(h^{\prime}_{2})$. It follows from
Lemma 3.3 in Section 3 that $\mathcal{J}\in C^{1}(\mathcal{W},\mathbb{R})$.

Our second result is as follows.

######  Theorem 1.3.

Let $G=(V,E)$ be a connected locally finite graph satisfying (3) and (4).
Suppose that $(h^{\prime}_{1})$ and $(h^{\prime}_{2})$ hold. Then the equation
(1) admits a sequence of solutions $\\{u_{n}\\}$ with
$\mathcal{J}(u_{n})\rightarrow+\infty$ as $n\rightarrow\infty$.

######  Remark 1.4.

_It should be pointed out that the hypotheses $(h^{\prime}_{1})$ and
$(h^{\prime}_{2})$ were proposed by Chang, Wang and Yang in [7], where they
got the existence of ground state solutions to the equation (1) on graphs.
Different from their results in [7], by using the symmetric mountain pass
theorem, we obtain infinitely many solutions with high energy of the equation
(1). On the other hand, we extend the classical results in Euclidean space to
discrete graphs._

This article is organized as follows. In Sect. 2, we prove the existence of
the nontrivial solutions of equation (1) from local to global. In Sect. 3, we
prove Theorem 1.3 by using the symmetric mountain pass theorem.

##  2 Existence of global solutions

In this section, we prove Theorem 1.1 by using variational method from local
to global. Firstly, we present some preliminaries and basic functional
settings. Let $\Omega$ be a connected finite subset of $V$. The boundary of
$\Omega$ in $V$ is defined by

$\partial\Omega:=\\{y\notin\Omega:\exists x\in\Omega\ \ \hbox{such that}\ \
xy\in E\\}.$ (8)

The space $\mathcal{H}(\Omega):=W^{1,2}_{0}(\Omega)$ endowed with the inner
product

$(u,v)_{\mathcal{H}(\Omega)}=\left(\int_{\overline{\Omega}}|\nabla
u|^{2}d\mu+\int_{\Omega}h(x)u^{2}d\mu\right)^{\frac{1}{2}}$

is a Hilbert space, where $\overline{\Omega}=\Omega\cup\partial\Omega$. Fixed
some point $O\in V$. For any $x\in V$, $\rho(x)=\rho(x,O)$ denotes the
distance between $x$ and $O$. For any integer $k\geq 1$, we denote a ball
centered at $O$ with radius $k$ by

$B_{k}=B_{k}(O)=\\{x\in V:\rho(x)<k\\}.$

The boundary of $B_{k}$ written as

$\partial B_{k}=\\{x\in V:\rho(x)=k\\}.$

It is easy to see that if $\Omega=B_{k}$, then $\partial\Omega$ defined in (8)
is equal to $\partial B_{k}$.

Now we consider the following locally equation

$\displaystyle\begin{cases}-\Delta u+h(x)u=u\log u^{2},&\text{in}\ \ B_{k},\\\
u=0,&\text{on}\ \ \partial B_{k},\end{cases}$ (9)

where $h$ satisfies $(h_{1})$ and $(h_{2})$. It is suitable to study (9) in
the space $\mathcal{H}(B_{k})$ under the norm

$\|u\|_{\mathcal{H}(B_{k})}=\left(\int_{\overline{B}_{k}}|\nabla
u|^{2}d\mu+\int_{B_{k}}h(x)u^{2}d\mu\right)^{\frac{1}{2}}.$

The functional related to (9) is

$\mathcal{I}_{k}(u)=\frac{1}{2}\|u\|^{2}_{\mathcal{H}(B_{k})}+\frac{1}{2}\int_{B_{k}}u^{2}d\mu-\frac{1}{2}\int_{B_{k}}u^{2}\log
u^{2}d\mu.$

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

$|t^{2}\log t^{2}|\leq
C_{\varepsilon}(|t|^{2-\varepsilon}+|t|^{2+\varepsilon}),\ \ \forall
t\in\mathbb{R}\backslash\\{0\\}.$ (10)

By (10) and a standard argument, we have $\mathcal{I}_{k}\in
C^{1}(\mathcal{H}(B_{k}),\mathbb{R})$, and

$\langle\mathcal{I}^{\prime}_{k}(u),v\rangle=\int_{\overline{B}_{k}}\nabla
u\nabla vd\mu+\int_{B_{k}}h(x)uvd\mu-\int_{B_{k}}u^{2}\log u^{2}d\mu.$

Next, we present a result about the compactness of the Sobolev space
$\mathcal{H}$ defined by (5).

######  Lemma 2.1.

Let $G=(V,E)$ be a connected locally finite graph satisfying (3) and (4).
Suppose that $(h_{1})$ and $(h_{2})$ hold. Then $\mathcal{H}$ is weakly pre-
compact and $\mathcal{H}$ is compactly embedded into $L^{q}(V)$ for any
$q\in[1,+\infty]$.

###### Proof.

The proof of this lemma is similar to Lemma 2.6 in [24], so we omit it. ∎

The following lemma gives the compactness of the space $\mathcal{H}(\Omega)$.
In addition, we prove an embedding inequality where the embedding constant is
independent of $\Omega$. For brevity, we denote the $L^{q}$ norms on $\Omega$
by $\|\cdot\|_{q,\Omega}$.

######  Lemma 2.2.

Let $G=(V,E)$ be a connected locally finite graph satisfying (3) and (4),
$\Omega$ be a connected finite subset of $V$. Assume that $(h_{1})$ hold. Then
$\mathcal{H}(\Omega)$ is compactly embedded into $L^{q}(\Omega)$ for any
$q\in[2,+\infty]$. In particular, there exists a constant $C$ depending only
on $h_{0}$, $\mu_{\min}$ and $q$ such that for any $u\in\mathcal{H}(\Omega)$,

$\|u\|_{q,\Omega}\leq C\|u\|_{\mathcal{H}(\Omega)}.$ (11)

Moreover, $\mathcal{H}(\Omega)$ is pre-compact.

###### Proof.

The proof of this lemma is inspired by Theorem 7 in [10]. Since $\Omega$ is a
connected finite subset of $V$, $\mathcal{H}(\Omega)$ is a finite dimensional
space. Hence, $\mathcal{H}(\Omega)$ is pre-compact. We are left to prove (11).
Indeed, for any $u\in\mathcal{H}(\Omega)$ and vertex $x_{0}\in\Omega$, we have

$\displaystyle\|u\|^{2}_{\mathcal{H}(\Omega)}$
$\displaystyle=\int_{\overline{\Omega}}|\nabla
u|^{2}d\mu+\int_{\Omega}h(x)u^{2}d\mu$
$\displaystyle\geq\int_{\Omega}h(x)u^{2}d\mu$
$\displaystyle=\underset{x\in\Omega}{\sum}h(x)u(x)^{2}\mu(x)$
$\displaystyle\geq h_{0}\mu_{\min}u(x_{0})^{2},$

which gives

$u(x_{0})\leq\left(\frac{1}{h_{0}\mu_{\min}}\right)^{\frac{1}{2}}\|u\|_{\mathcal{H}(\Omega)}.$

Then

$\|u\|_{\infty,\Omega}\leq\left(\frac{1}{h_{0}\mu_{\min}}\right)^{\frac{1}{2}}\|u\|_{\mathcal{H}(\Omega)},$

and thus for any $2\leq q<+\infty$,

$\displaystyle\int_{\Omega}|u|^{q}d\mu$
$\displaystyle=\int_{\Omega}|u|^{2}|u|^{q-2}d\mu$
$\displaystyle\leq\left(\frac{1}{h_{0}\mu_{\min}}\right)^{\frac{q-2}{2}}\|u\|^{q-2}_{\mathcal{H}(\Omega)}\int_{\Omega}u^{2}d\mu$
$\displaystyle\leq(h_{0})^{\frac{-q}{2}}(\mu_{\min})^{\frac{2-q}{2}}\|u\|^{q-2}_{\mathcal{H}(\Omega)}\int_{\Omega}h(x)u^{2}d\mu$
$\displaystyle\leq(h_{0})^{\frac{-q}{2}}(\mu_{\min})^{\frac{2-q}{2}}\|u\|^{q}_{\mathcal{H}(\Omega)}.$

Therefore, for any $2\leq q\leq+\infty$,

$\|u\|_{q,\Omega}\leq C\|u\|_{H(\Omega)},$

where $C$ depends only on $h_{0}$, $\mu_{\min}$ and $q$. ∎

Now, we list two lemmas about Green’s formulas on graphs, which are
fundamental when we use methods in calculus of variations.

######  Lemma 2.3 ([24, Lemma 2.1]).

Let $G=(V,E)$ be a connected locally finite graph satisfying (3) and (4). Then
for any $v\in C_{c}(V)$, we have

$\int_{V}\nabla u\nabla vd\mu=-\int_{V}(\Delta u)vd\mu,$

where $C_{c}(V)$ is the set of all functions with finite supports in $V$.

######  Lemma 2.4 ([24, Lemma 2.2]).

Let $G=(V,E)$ be a connected locally finite graph satisfying (3) and (4),
$\Omega$ be a connected finite subset of $V$ and $u\in W_{0}^{1,2}(\Omega)$.
Then for any $v\in C_{c}(\Omega)$, we have

$\int_{\overline{\Omega}}\nabla u\nabla vd\mu=-\int_{\Omega}(\Delta u)vd\mu,$

where $C_{c}(\Omega)$ denotes the set of all functions
$u:\Omega\rightarrow\mathbb{R}$ satisfying $\emph{\hbox{supp}}\
u\subset\Omega$ and $u=0$ on $\partial\Omega$.

Next, we verify that the functional $\mathcal{I}_{k}$ satisfies the mountain
pass geometry in the following two lemmas.

######  Lemma 2.5.

There exist constants $\delta>0$ and $r>0$ satisfying
$\mathcal{I}_{k}(u)\geq\delta$ for all $u$ with
$\|u\|_{\mathcal{H}(B_{k})}=r$.

###### Proof.

It follows from (10) and Lemma 2.2 that

$\displaystyle\mathcal{I}_{k}(u)$
$\displaystyle=\frac{1}{2}\|u\|^{2}_{\mathcal{H}(B_{k})}+\frac{1}{2}\int_{B_{k}}u^{2}d\mu-\frac{1}{2}\int_{B_{k}}u^{2}\log
u^{2}d\mu$
$\displaystyle\geq\frac{1}{2}\|u\|^{2}_{\mathcal{H}(B_{k})}-\frac{1}{2}\int_{\\{x\in
B_{k}:u(x)\geq 1\\}}u^{2}\log u^{2}d\mu$
$\displaystyle\geq\frac{1}{2}\|u\|^{2}_{\mathcal{H}(B_{k})}-\frac{1}{2}\int_{B_{k}}C_{\varepsilon}|u|^{2+\varepsilon}d\mu$
$\displaystyle\geq\frac{1}{2}\|u\|^{2}_{\mathcal{H}(B_{k})}-\frac{C_{\varepsilon}}{2}\|u\|^{2+\varepsilon}_{2+\varepsilon,B_{k}}$
$\displaystyle\geq\frac{1}{2}\|u\|^{2}_{\mathcal{H}(B_{k})}-\frac{C_{\varepsilon}}{2}C\|u\|^{2+\varepsilon}_{\mathcal{H}(B_{k})}$
$\displaystyle=\frac{1}{2}\|u\|^{2}_{\mathcal{H}(B_{k})}\left(1-C_{\varepsilon}C\|u\|^{\varepsilon}_{\mathcal{H}(B_{k})}\right),$

where $C$ depends on $h_{0}$, $\mu_{\min}$ and $\varepsilon$. Setting
$r=\left(\frac{1}{2C_{\varepsilon}C}\right)^{\frac{1}{\varepsilon}}$, we have
$\mathcal{I}_{k}(u)\geq\frac{1}{4}r^{2}$ for all $u$ with
$\|u\|_{\mathcal{H}(B_{k})}=r$. This gives the desired result. ∎

######  Lemma 2.6.

There exists some $u\in\mathcal{H}(B_{k})$ with $u(x)\geq 0$ for all $x\in
B_{k}$ such that $\mathcal{I}_{k}(tu)\rightarrow-\infty$ as
$t\rightarrow+\infty$.

###### Proof.

For any fixed point $x_{0}\in B_{k}$, we set

$u(x)=\left\\{\begin{array}[]{lll}1&{\rm when}&x=x_{0}\\\\[6.45831pt] 0&{\rm
when}&x\not=x_{0}.\end{array}\right.$

Then we have

$\mathcal{I}_{k}(tu)=\frac{t^{2}}{2}\left(\sum_{y\sim x_{0}}\mu(y)|\nabla
u|^{2}(y)+\mu(x_{0})|\nabla
u|^{2}(x_{0})\right)+\frac{t^{2}}{2}\mu(x_{0})(h(x_{0})+1)-\frac{t^{2}}{2}\log
t^{2}\mu(x_{0})\rightarrow-\infty$

as $t\rightarrow+\infty$. ∎

Next, we prove that $\mathcal{I}_{k}$ satisfies the $(PS)_{c}$ condition for
any $c\in\mathbb{R}$.

######  Lemma 2.7.

For any $c\in\mathbb{R}$, if $\\{u_{n}\\}\subset\mathcal{H}(B_{k})$ satisfies
$\mathcal{I}_{k}(u_{n})\rightarrow c$ and
$\mathcal{I}_{k}^{\prime}(u_{n})\rightarrow 0$, then there exists some
$u\in\mathcal{H}(B_{k})$ such that, up to a subsequence, $u_{n}\rightarrow u$
in $\mathcal{H}(B_{k})$.

###### Proof.

Note that $\\{u_{n}\\}\subset\mathcal{H}(B_{k})$ satisfies

$\mathcal{I}_{k}(u_{n})\rightarrow c\ \ \hbox{and}\ \
\mathcal{I}_{k}^{\prime}(u_{n})\rightarrow 0\ \ \hbox{as}\ \
n\rightarrow\infty.$ (12)

We claim that $\\{u_{n}\\}$ is bounded in $\mathcal{H}(B_{k})$. In fact, by
(12) we have

$\underset{n\rightarrow\infty}{\lim}\mathcal{I}_{k}(u_{n})=\underset{n\rightarrow\infty}{\lim}\left(\mathcal{I}_{k}(u_{n})-\frac{1}{2}\langle\mathcal{I}^{\prime}_{k}(u_{n}),u_{n}\rangle\right)=\frac{1}{2}\underset{n\rightarrow\infty}{\lim}\int_{B_{k}}u^{2}_{n}d\mu=\frac{1}{2}\underset{n\rightarrow\infty}{\lim}\|u_{n}\|^{2}_{2,B_{k}}=c.$
(13)

On the other hand, by the Hölder’s inequality, for any $\varepsilon\in(0,1)$,
there exist $C_{\varepsilon},C_{1}>0$ such that

$\displaystyle\int_{B_{k}}u^{2}_{n}\log u_{n}^{2}d\mu$
$\displaystyle\leq\int_{\\{x\in B_{k}:u_{n}(x)\geq 1\\}}u^{2}_{n}\log
u_{n}^{2}d\mu$ $\displaystyle\leq
C_{\varepsilon}\int_{B_{k}}|u_{n}|^{2+\varepsilon}d\mu$ $\displaystyle\leq
C_{\varepsilon}\left(\int_{B_{k}}|u_{n}|^{2}d\mu\right)^{\frac{1}{2}}\left(\int_{B_{k}}|u_{n}|^{2(1+\varepsilon)}d\mu\right)^{\frac{1}{2}}$
$\displaystyle=C_{\varepsilon}\|u_{n}\|_{2,B_{k}}\|u_{n}\|^{1+\varepsilon}_{2(1+\varepsilon),B_{k}}$
$\displaystyle\leq
C_{\varepsilon}C_{1}\|u_{n}\|_{2,B_{k}}\|u_{n}\|^{1+\varepsilon}_{\mathcal{H}(B_{k})}$
$\displaystyle\leq\frac{1+\varepsilon}{2}\|u_{n}\|^{2}_{\mathcal{H}(B_{k})}+\frac{1-\varepsilon}{2}C_{\varepsilon}C_{1}\|u_{n}\|^{\frac{2}{1-\varepsilon}}_{2,B_{k}},$
(14)

where $C_{1}$ depends on $h_{0}$, $\mu_{\min}$, $\varepsilon$ and the last
inequality is derived from the Young inequality. Since
$\langle\mathcal{I}^{\prime}_{k}(u_{n}),u_{n}\rangle=0$, combining (13) and
(2), we obtain

$\|u_{n}\|^{2}_{\mathcal{H}(B_{k})}=\int_{B_{k}}u^{2}_{n}\log
u^{2}_{n}d\mu\leq\frac{1+\varepsilon}{2}\|u_{n}\|^{2}_{\mathcal{H}(B_{k})}+\frac{1-\varepsilon}{2}C_{\varepsilon}C_{1}\|u_{n}\|^{\frac{2}{1-\varepsilon}}_{2,B_{k}},$
(15)

which implies that $\\{u_{n}\\}$ is bounded in $\mathcal{H}(B_{k})$, and thus
there exists a constant $M$ independent of $k$ such that
$\|u_{n}\|_{\infty,B_{k}}\leq M(c,\varepsilon,\mu_{\min},h_{0})$. By Lemma
2.2, up to a subsequence, there exists some $u\in\mathcal{H}(B_{k})$ such that
$u_{n}\rightarrow u$ in $\mathcal{H}(B_{k})$ as $n\rightarrow\infty$. ∎

Now, we will prove that the local equation (9) admits a nontrivial solution.

######  Lemma 2.8.

Let $G=(V,E)$ be a connected locally finite graph satisfying (3) and (4).
Suppose that $(h_{1})$ hold. Then equation (9) has a nontrivial solution.

###### Proof.

By Lemma 2.5, Lemma 2.6 and Lemma 2.7, $\mathcal{I}_{k}$ satisfies all the
hypotheses of the mountain-pass theorem. Using the mountain-pass theorem due
to Ambrosetti-Rabinowitz [22], we conclude that

$c_{k}=\min_{\gamma\in\Gamma}\max_{u\in\gamma}\mathcal{I}_{k}(u)$

is the critical level of $\mathcal{I}_{k}$, where

$\Gamma=\left\\{\gamma\in
C([0,1],\mathcal{H}(B_{k})):\gamma(0)=0,\gamma(1)=e\right\\}.$

In particular, there exists some $u_{k}\in\mathcal{H}(B_{k})$ such that
$\mathcal{I}_{k}(u_{k})=c_{k}\geq\delta>0$ and $u_{k}\not\equiv 0$ satisfies
the Euler-Lagrange equation

$\displaystyle\begin{cases}-\Delta u_{k}+h(x)u_{k}=u^{2}_{k}\log
u_{k}^{2},&\text{in}\ \ B_{k},\\\ u_{k}=0,&\text{on}\ \ \partial
B_{k}.\end{cases}$ (16)

∎

At the end of this section, we prove the existence of global solutions to the
equation (1).

_The proof of Theorem 1.1_. We will divide the proof into two steps as
follows.

(i) For any finite set $A\subset V$, the nontrivial solution $u_{k}$ of (9) is
uniformly bounded in $A$.

Let $u_{k}$ be a nontrivial solution of (9). Similar to the proof of (15) in
Lemma 2.7, we have

$\|u_{k}\|^{2}_{\mathcal{H}(B_{k})}=\int_{\overline{B}_{k}}|\nabla
u_{k}|^{2}d\mu+\int_{B_{k}}h(x)u^{2}_{k}d\mu\leq C$ (17)

for some constant $C$ independent of $k$. Note that for any finite set
$A\subset V$, there exists $B_{k}$ such that $A\subset B_{k}$ for sufficiently
large $k$. Since

$\displaystyle\|u_{k}\|^{2}_{\mathcal{H}(B_{k})}$
$\displaystyle=\int_{\overline{B}_{k}}|\nabla
u_{k}|^{2}d\mu+\int_{B_{k}}h(x)u^{2}_{k}d\mu$
$\displaystyle\geq\int_{B_{k}}h(x)u^{2}_{k}d\mu$ $\displaystyle\geq
h_{0}\int_{A}u^{2}_{k}d\mu$ $\displaystyle\geq
h_{0}\mu_{\min}u^{2}_{k}(x_{0})$

for any fixed $x_{0}\in A$, which ensures that

$\|u_{k}\|_{\infty,A}\leq\left(\frac{1}{h_{0}\mu_{\min}}\right)^{\frac{1}{2}}\|u_{k}\|_{\mathcal{H}(B_{k})}\leq\sqrt{\frac{C}{h_{0}\mu_{\min}}}.$

Thus $u_{k}$ is uniformly bounded in $A$.

(ii) The equation (1) admits a nontrivial solution in $\mathcal{H}$.

Note that $u_{k}$ is naturally viewed as a sequence of functions defined on
$V$, say $u_{k}\equiv 0$ on $V\backslash B_{k}$. Then by the step (i), there
would exist a subsequence of $\\{u_{k}\\}$, which is still denoted by
$\\{u_{k}\\}$, and a function $u^{*}$ such that $u_{k}$ converges to $u^{*}$
locally uniformly in $V$, i.e. for any fixed positive integer $l$,

$\underset{k\rightarrow\infty}{\lim}u_{k}(x)=u^{*},\ \ \forall x\in B_{l}.$

Next we prove that $u^{*}\in\mathcal{H}$. Since $u_{k}\equiv 0$ on
$V\backslash B_{k}$, by (17) we have

$\displaystyle\|u_{k}\|^{2}_{\mathcal{H}}$ $\displaystyle=\int_{V}(|\nabla
u_{k}|^{2}+h(x)u^{2}_{k})d\mu$ $\displaystyle=\underset{x\in V}{\sum}|\nabla
u_{k}|^{2}(x)\mu(x)+\underset{x\in B_{k}}{\sum}h(x)u^{2}_{k}(x)\mu(x)$
$\displaystyle=\frac{1}{2}\underset{x\in B_{k}}{\sum}\underset{y\sim
x}{\sum}\omega_{xy}(u_{k}(y)-u_{k}(x))^{2}+\underset{x\in
B_{k}}{\sum}h(x)u^{2}_{k}(x)\mu(x)+\frac{1}{2}\underset{x\in\partial
B_{k}}{\sum}\underset{y\sim x}{\sum}\omega_{xy}(u_{k}(y)-u_{k}(x))^{2}$
$\displaystyle\leq 2\|u_{k}\|^{2}_{\mathcal{H}(B_{k})}$ $\displaystyle\leq
2C,$

which implies that $\\{u_{k}\\}$ is bounded in $\mathcal{H}$. Since
$\mathcal{H}$ is weakly compact, it follows that up to a subsequence,
$\\{u_{k}\\}$ converges to some function $u^{*}_{1}$ weakly in $\mathcal{H}$.
This in particular implies

$\int_{V}u_{k}\phi d\mu\rightarrow\int_{V}u^{*}_{1}\phi d\mu,\ \
\forall\phi\in C_{c}(V).$

For any fixed point $z\in V$, taking

$\displaystyle\phi(x)=\begin{cases}1,&\text{if}\ \ x=z,\\\ 0,&\text{if}\ \
x\neq z.\end{cases}$

Then $u_{k}(z)\rightarrow u^{*}_{1}(z).$ Hence by the uniqueness of the limit,
$u^{*}_{1}(z)=u^{*}(z)$ for all $z\in V$ and then $u^{*}\in\mathcal{H}$. Thus,
it follows from (16) that for any fixed $x\in V$, there hold

$-\Delta u^{*}+h(x)u^{*}=u^{*}\log(u^{*})^{2}.$

Therefore $u^{*}$ is a solution of (1).

Finally, we prove that $u^{*}$ is nontrivial. Since $u_{k}$ is a nontrivial
solution of equation (9),
$\langle\mathcal{I}_{k}^{\prime}(u_{k}),u_{k}\rangle=0$. By (10) and Lemma
2.2, for any $0<\varepsilon<1$, there exists $C_{\varepsilon}$ such that

$\displaystyle\|u_{k}\|^{2}_{\mathcal{H}(B_{k})}$
$\displaystyle=\int_{B_{k}}u_{k}^{2}\log u^{2}_{k}d\mu$
$\displaystyle\leq\int_{\\{x\in B_{k}:u_{k}(x)\geq 1\\}}u_{k}^{2}\log
u^{2}_{k}d\mu$ $\displaystyle\leq
C_{\varepsilon}\int_{B_{k}}|u_{k}|^{2+\varepsilon}d\mu$ $\displaystyle\leq
C_{\varepsilon}C_{2}\|u_{k}\|^{2+\varepsilon}_{\mathcal{H}(B_{k})}.$

Thus, we obtain

$\|u_{k}\|^{2}_{\mathcal{H}(B_{k})}\geq(C_{\varepsilon}C_{2})^{-\frac{2}{\varepsilon}}>0,$

where $C_{\varepsilon}$ and $C_{2}$ is independent of $k$. Hence it implies
that there exists a constant $\theta$ independent of $k$ satisfying
$0<\theta<(C_{\varepsilon}C_{2})^{-\frac{2}{\varepsilon}}$ such that

$\int_{V}u_{k}^{2}\log u^{2}_{k}d\mu\geq\int_{B_{k}}u_{k}^{2}\log
u^{2}_{k}d\mu=\|u_{k}\|^{2}_{\mathcal{H}(B_{k})}>\theta>0.$ (18)

Note that $u_{k}\rightharpoonup u^{*}$ in $\mathcal{H}$. Then it follows from
(10) that, for any $0<\varepsilon<1$, there exists $C_{\varepsilon}$ such that

$\int_{V}|u_{k}^{2}\log u^{2}_{k}|d\mu\leq
C_{\varepsilon}\int_{V}(|u_{k}|^{2-\varepsilon}+|u_{k}|^{2+\varepsilon})d\mu.$

Thus by Lemma 2.2, Lebesgue dominated convergence theorem and (18), we have

$\int_{V}(u^{*})^{2}\log(u^{*})^{2}d\mu=\underset{k\rightarrow\infty}{\lim}\int_{V}u_{k}^{2}\log
u^{2}_{k}d\mu\geq\theta>0,$

which implies that there must exist a vertex $x_{0}\in V$ such that
$\log(u^{*}(x_{0}))^{2}>0$ and consequently there holds $|u^{*}(x_{0})|>1$.
Therefore, $u^{*}\not\equiv 0$ and this completes the proof of the theorem.

##  3 Existence of multiple solutions

In this section, we will prove that the equation (1) admits infinitely many
solutions by using the following symmetric mountain pass theorem.

######  Lemma 3.1 ([18, Theorem 9.12]).

Let $X$ be an infinite dimensional Banach space, and let
$J\in\mathcal{C}^{1}(X,\mathbb{R})$ be even, satisfying $(PS)$ condition and
$J(0)=0$. If $X=Y\bigoplus Z$, where $Y$ is finite dimensional and $J$
satisfies

(i) there are constant $\varrho,\sigma>0$ such that $J|_{\partial
B_{\varrho}\cap Z}\geq\sigma$ and

(ii) for each finite dimensional subspace $\widetilde{X}\subset X$ there
exists an $R=R(\widetilde{X})$ such that $J\leq 0$ on $\widetilde{X}\setminus
B_{R(\widetilde{X})}$,

then $J$ possesses an unbounded sequence of critical values.

Now, we present a Sobolev embedding result, by which we know that the
functional $\mathcal{J}$ defined by (6) is well defined in $\mathcal{W}$,
where $\mathcal{W}$ is defined by (7).

######  Lemma 3.2 ([7, Lemma 4]).

Let $G=(V,E)$ be a connected locally finite graph satisfying (3) and (4).
Assume that $(h^{\prime}_{1})$ and $(h^{\prime}_{2})$ hold. Then $\mathcal{W}$
is weakly pre-compact and $\mathcal{W}$ is compactly embedded into $L^{q}(V)$
for any $q\in[1,+\infty]$.

The following lemma gives the same conclusion as Proposition 9 in [7], which
tells us that the functional $\mathcal{J}\in C^{1}(\mathcal{W},\mathbb{R})$,
while the authors in [7] did not give a detailed proof of it. We supplement
the proof here for the convenience of the readers.

######  Lemma 3.3.

Under the same assumptions as in Lemma 3.2, the functional $\mathcal{J}\in
C^{1}(\mathcal{W},\mathbb{R})$. Furthermore, for any $v\in\mathcal{W}$,

$\langle\mathcal{J}^{\prime}(u),v\rangle=\int_{V}(\nabla u\nabla
v+h(x)uv)d\mu-\int_{V}uv\log u^{2}d\mu.$ (19)

###### Proof.

For any $u\in\mathcal{W}$, let

$\varphi(u)=\frac{1}{2}\int_{V}(|\nabla u|^{2}+(h(x)+1)u^{2})d\mu$

and

$\psi(u)=\frac{1}{2}\int_{V}u^{2}\log u^{2}d\mu.$

Then $\mathcal{J}(u)=\varphi(u)-\psi(u)$. We only need to prove that
$\varphi,\psi\in C^{1}(\mathcal{W},\mathbb{R})$ and for any $v\in\mathcal{W}$,

$\langle\varphi^{\prime}(u),v\rangle=\int_{V}(\nabla u\nabla
v+(h(x)+1)uv)d\mu,$ $\langle\psi^{\prime}(u),v\rangle=\int_{V}(uv+uv\log
u^{2})d\mu.$

First, we verify $\varphi\in C^{1}(\mathcal{W},\mathbb{R})$. Indeed, for any
$v\in\mathcal{W}$,

$\displaystyle(\varphi^{\prime}(u),v)$ $\displaystyle=\underset{t\rightarrow
0}{\lim}\frac{\varphi(u+tv)-\varphi(u)}{t}$
$\displaystyle=\frac{1}{2}\underset{t\rightarrow
0}{\lim}\frac{1}{t}\left[\int_{V}(|\nabla(u+tv)|^{2}+(h(x)+1)(u+tv)^{2})d\mu-\int_{V}(|\nabla
u|^{2}+(h(x)+1)u^{2})d\mu\right]$ $\displaystyle=\int_{V}(\nabla u\nabla
v+(h(x)+1)uv)d\mu.$

Let $u_{n}\rightarrow u$ in $\mathcal{W}$, then $u_{n}\rightarrow u$ in
$L^{2}(V)$. We claim that
$\|\varphi^{\prime}(u_{n})-\varphi^{\prime}(u)\|_{\mathcal{W}^{*}}\rightarrow
0$ as $n\rightarrow\infty$, where $\mathcal{W}^{*}$ is the dual space of
$\mathcal{W}$. In fact, for any $v\in\mathcal{W}$ with
$\|v\|_{\mathcal{W}}=1$, we have

$\displaystyle\langle\varphi^{\prime}(u_{n})-\varphi^{\prime}(u),v\rangle$
$\displaystyle=\int_{V}[\nabla(u_{n}-u)\nabla v+(h(x)+1)(u_{n}-u)v]d\mu$
$\displaystyle=\int_{V}[\nabla(u_{n}-u)\nabla
v+h(x)(u_{n}-u)v]d\mu+\int_{V}(u_{n}-u)vd\mu$
$\displaystyle\leq\left(\int_{V}|\nabla(u_{n}-u)|^{2}d\mu\right)^{\frac{1}{2}}\left(\int_{V}|\nabla
v|^{2}d\mu\right)^{\frac{1}{2}}$
$\displaystyle\phantom{{}={}}+\left(\int_{V}h(x)(u_{n}-u)^{2}d\mu\right)^{\frac{1}{2}}\left(\int_{V}h(x)v^{2}d\mu\right)^{\frac{1}{2}}+\left(\int_{V}(u_{n}-u)^{2}d\mu\right)^{\frac{1}{2}}\left(\int_{V}v^{2}d\mu\right)^{\frac{1}{2}}$
$\displaystyle\leq
2\|u_{n}-u\|_{\mathcal{W}}\|v\|_{\mathcal{W}}+\|u_{n}-u\|_{2}\|v\|_{2}\rightarrow
0\ \ \hbox{as}\ \ n\rightarrow\infty.$

Hence,
$\|\varphi^{\prime}(u_{n})-\varphi^{\prime}(u)\|_{\mathcal{W}^{*}}=\underset{\|v\|_{\mathcal{W}}=1}{\sup}\langle\varphi^{\prime}(u_{n})-\varphi^{\prime}(u),v\rangle\rightarrow
0$ as $n\rightarrow\infty$. Therefore, $\varphi\in
C^{1}(\mathcal{W},\mathbb{R})$.

Next, we prove that $\psi\in C^{1}(\mathcal{W},\mathbb{R})$. Let
$u,v\in\mathcal{W}\hookrightarrow L^{q}(V)$, $q\geq 1$, then we have $u,v\in
L^{q}(V)$. Given $x\in V$ and $0<|t|<1$, by the mean value theorem and (10),
there exists $\xi\in(0,1)$ such that

$\displaystyle\frac{\left|(u+tv)^{2}\log(u+tv)^{2}-u^{2}\log
u^{2}\right|}{|t|}$ $\displaystyle=$ $\displaystyle 2\left|(u+\xi tv)\cdot
v\cdot\log(u+\xi tv)^{2}+(u+\xi tv)\cdot v\right|$ $\displaystyle\leq$
$\displaystyle 2C_{\varepsilon}(|u+\xi tv|^{1-\varepsilon}+|u+\xi
tv|^{1+\varepsilon})|v|+2|u+\xi tv|\cdot|v|$ $\displaystyle\leq$
$\displaystyle
2^{2-\varepsilon}C_{\varepsilon}(|u|^{1-\varepsilon}+|v|^{1-\varepsilon})|v|+2^{2+\varepsilon}C_{\varepsilon}(|u|^{1+\varepsilon}+|v|^{1+\varepsilon})|v|+2|u+v|\cdot|v|.$

The Hölder’s inequality implies that

$\left[2^{2-\varepsilon}C_{\varepsilon}(|u|^{1-\varepsilon}+|v|^{1-\varepsilon})|v|+2^{2+\varepsilon}C_{\varepsilon}(|u|^{1+\varepsilon}+|v|^{1+\varepsilon})|v|+2|u+v|\cdot|v|\right]\in
L^{1}(V).$

It follows from Lebesgue dominated convergence theorem that

$\displaystyle\langle\psi^{\prime}(u),v\rangle$
$\displaystyle=\underset{t\rightarrow 0}{\lim}\frac{\psi(u+tv)-\psi(u)}{t}$
$\displaystyle=\frac{1}{2}\int_{V}\underset{t\rightarrow
0}{\lim}\frac{(u+tv)^{2}\log(u+tv)^{2}-u^{2}\log u^{2}}{t}d\mu$
$\displaystyle=\int_{V}(uv+uv\log u^{2})d\mu.$

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

$|u\log u^{2}|\leq
C_{\varepsilon}(|u|^{1-\varepsilon}+|u|^{1+\varepsilon})=C_{\varepsilon}\left(|u|^{\frac{2-\varepsilon}{(2-\varepsilon)/(1-\varepsilon)}}+|u|^{\frac{2+\varepsilon}{(2+\varepsilon)/(1+\varepsilon)}}\right).$

To prove the Gâteaux derivative of $\psi$ is continuous, we let $f(u)=u\log
u^{2}$ and assume that $u_{n}\rightarrow u$ in $\mathcal{W}$ as
$n\rightarrow\infty$. Then by Lemma 3.2, up to a subsequence, we have
$u_{n}\rightarrow u$ in $L^{2-\varepsilon}(V)\cap L^{2+\varepsilon}(V)$, where
$||\cdot||_{L^{2-\varepsilon}(V)\cap
L^{2+\varepsilon}(V)}:=||\cdot||_{{2-\varepsilon}}+||\cdot||_{2+\varepsilon}$.
It follows from Lemma 5.12 in [14] that

$f(u_{n})\rightarrow f(u)\ \ \hbox{in}\ \
L^{\frac{2-\varepsilon}{1-\varepsilon}}(V)+L^{\frac{2+\varepsilon}{1+\varepsilon}}(V).$

Here, for any $w\in L^{r}(V)+L^{s}(V),r,s\geq 1$, we define the norm

$\|w\|_{L^{r}(V)+L^{s}(V)}:=\inf\\{||w_{1}||_{r}+||w_{2}||_{s}:w_{1}\in
L^{r}(V),w_{2}\in L^{s}(V),w=w_{1}+w_{2}\\}.$

On the other hand, by the Hölder’s inequality, we obtain

$\displaystyle|\langle\psi^{\prime}(u_{n})-\psi^{\prime}(u),v\rangle|$
$\displaystyle=\left|\int_{V}(u_{n}\log u^{2}_{n}-u\log
u^{2})vd\mu+\int_{V}(u_{n}-u)vd\mu\right|$
$\displaystyle\leq\|f(u_{n})-f(u)\|_{L^{\frac{2-\varepsilon}{1-\varepsilon}}(V)+L^{\frac{2+\varepsilon}{1+\varepsilon}}(V)}\|v\|_{L^{2-\varepsilon}(V)\cap
L^{2+\varepsilon}(V)}+\|u_{n}-u\|_{2}\|v\|_{2}.$

Then

$\|\psi^{\prime}(u_{n})-\psi^{\prime}(u)\|_{\mathcal{W}^{*}}\rightarrow 0\ \
\hbox{as}\ \ n\rightarrow\infty,$

which implies that $\psi\in C^{1}(\mathcal{W},\mathbb{R})$. Thus
$\mathcal{J}\in C^{1}(\mathcal{W},\mathbb{R})$ and (19) holds. ∎

Next, we verify that the functional $\mathcal{J}$ satisfies the $(PS)$
condition.

######  Lemma 3.4.

Under the same assumptions as in Lemma 3.2. Then for any $c\in\mathbb{R}$,
$\mathcal{J}$ satisfies the $(PS)_{c}$ condition.

###### Proof.

Assume that $\\{u_{n}\\}\subset\mathcal{W}$ satisfies
$\mathcal{J}(u_{n})\rightarrow c$ and $\mathcal{J}^{\prime}(u_{n})\rightarrow
0$ as $n\rightarrow\infty$. Then we obtain

$\|u_{n}\|_{2}=\int_{V}u_{n}^{2}d\mu=2\mathcal{J}(u_{n})-\langle\mathcal{J}^{\prime}(u_{n}),u_{n}\rangle\leq
c+1+o(1)\|u_{n}\|_{\mathcal{W}}.$ (20)

Similar to the proof of (2), by Lemma 3.2, Hölder’s inequality and Yong
inequality , we have for any $\varepsilon\in(0,1)$, there exist
$C_{\varepsilon},C_{3}>0$ such that

$\displaystyle\int_{V}u^{2}_{n}\log u_{n}^{2}d\mu$
$\displaystyle\leq\int_{\\{x\in V:u\geq 1\\}}u^{2}_{n}\log u_{n}^{2}d\mu$
$\displaystyle\leq C_{\varepsilon}\int_{V}|u_{n}|^{2+\varepsilon}d\mu$
$\displaystyle=C_{\varepsilon}\|u_{n}\|_{2}\|u_{n}\|^{1+\varepsilon}_{2(1+\varepsilon)}$
(21) $\displaystyle\leq
C_{\varepsilon}C_{3}\|u_{n}\|_{2}\|u_{n}\|^{1+\varepsilon}_{\mathcal{W}}$
$\displaystyle\leq\frac{1+\varepsilon}{2}\|u_{n}\|^{2}_{\mathcal{W}}+\frac{1-\varepsilon}{2}C^{\prime}_{\varepsilon}C^{\prime}_{3}\|u_{n}\|^{\frac{2}{1-\varepsilon}}_{2}.$

Note that $\langle\mathcal{J}^{\prime}(u_{n}),u_{n}\rangle=0$, then by (20)
and (3), we obtain

$\displaystyle\|u_{n}\|^{2}_{\mathcal{W}}$
$\displaystyle=\int_{V}u^{2}_{n}\log u_{n}^{2}d\mu+\|u_{n}\|^{2}_{2}$
$\displaystyle\leq\frac{1+\varepsilon}{2}\|u_{n}\|^{2}_{\mathcal{W}}+\frac{1-\varepsilon}{2}C^{\prime}_{\varepsilon}C^{\prime}_{3}\|u_{n}\|^{\frac{2}{1-\varepsilon}}_{2}+\|u_{n}\|^{2}_{2},$

which implies that $\\{u_{n}\\}$ is bounded in $\mathcal{W}$. Since
$\mathcal{W}$ is weakly pre-compact, there exists $u\in\mathcal{W}$ such that,
up to a subsequence,

$\left\\{\begin{aligned} &u_{n}\rightharpoonup u&\hbox{in}\ \ \mathcal{W},\\\
&u_{n}(x)\rightarrow u(x)&\forall x\in V,\\\ &u_{n}\rightarrow u&\hbox{in}\ \
L^{q}(V),\ \ q\in[1,+\infty].\end{aligned}\right.$ (22)

By direct calculation, one gets

$\displaystyle\|u_{n}-u\|^{2}_{\mathcal{W}}$
$\displaystyle=\langle\mathcal{J}^{\prime}(u_{n}),u_{n}-u\rangle-\langle\mathcal{J}^{\prime}(u),u_{n}-u\rangle+\|u_{n}-u\|^{2}_{2}$
$\displaystyle+\int_{V}u_{n}(u_{n}-u)\log u^{2}_{n}d\mu-\int_{V}u(u_{n}-u)\log
u^{2}d\mu$ (23)

It is easy to prove that

$\langle\mathcal{J}^{\prime}(u_{n}),u_{n}-u\rangle-\langle\mathcal{J}^{\prime}(u),u_{n}-u\rangle\rightarrow
0\ \ \hbox{and}\ \ \|u_{n}-u\|^{2}_{2}\rightarrow 0\ \ \hbox{as}\ \
n\rightarrow\infty$ (24)

by using (22). It follows from (10), (22) and the Hölder’s inequality that

$\displaystyle\left|\int_{V}u_{n}(u_{n}-u)\log u^{2}_{n}d\mu\right|$
$\displaystyle\leq\int_{V}|u_{n}(u_{n}-u)\log u^{2}_{n}|d\mu$
$\displaystyle\leq\int_{V}C_{\varepsilon}|u_{n}-u|(|u_{n}|^{1-\varepsilon}+|u_{n}|^{1+\varepsilon})d\mu$
(25) $\displaystyle\leq
C_{\varepsilon}(\|u_{n}-u\|_{2}\|u_{n}\|^{1-\varepsilon}_{2(1-\varepsilon)}+\|u_{n}-u\|_{2}\|u_{n}\|^{1+\varepsilon}_{2(1+\varepsilon)})$
$\displaystyle\rightarrow 0\ \ \hbox{as}\ \ n\rightarrow\infty.$

Similarly, we have

$\int_{V}u(u_{n}-u)\log u^{2}d\mu\rightarrow 0\ \ \hbox{as}\ \
n\rightarrow\infty.$ (26)

Thus, combining (3),(24), (3) and (26), we obtain

$\|u_{n}-u\|^{2}_{\mathcal{W}}\rightarrow 0\ \ \hbox{as}\ \
n\rightarrow\infty.$

Then we complete the proof. ∎

_The Proof of Theorem 1.3._ Let $X=\mathcal{W}$ in Lemma 3.1. Next, we prove
that the functional $\mathcal{J}$ satisfies conditions $(i)$ and $(ii)$ in
Lemma 3.1.

$(i)$ By (10) and Lemma 3.2, for any $0<\varepsilon<1$, there exists
$C_{\varepsilon}>0$ such that

$\displaystyle\mathcal{J}(u)$
$\displaystyle=\frac{1}{2}\|u\|^{2}_{\mathcal{W}}-\frac{1}{2}\int_{V}u^{2}\log
u^{2}d\mu$
$\displaystyle\geq\frac{1}{2}\|u\|^{2}_{\mathcal{W}}-\frac{1}{2}\int_{\\{x\in
V:u\geq 1\\}}u^{2}\log u^{2}d\mu$
$\displaystyle\geq\frac{1}{2}\|u\|^{2}_{\mathcal{W}}-\frac{1}{2}C_{\varepsilon}\|u\|^{2+\varepsilon}_{2+\varepsilon}$
$\displaystyle\geq\frac{1}{2}\|u\|^{2}_{\mathcal{W}}-\frac{1}{2}C_{\varepsilon}C_{4}\|u\|^{2+\varepsilon}_{\mathcal{W}}.$

Taking
$\varrho=\left(\frac{1}{2C_{\varepsilon}C_{4}}\right)^{\frac{1}{\varepsilon}}$,
then we have

$\mathcal{J}(u)|_{\partial B_{\varrho}}\geq\frac{1}{4}\varrho^{2}:=\sigma>0.$

$(ii)$ Let $\widetilde{\mathcal{W}}$ be a finite dimensional subspace of
$\mathcal{W}$. Suppose $u\in\widetilde{\mathcal{W}}\setminus\\{0\\}$ and
setting $\phi:=\frac{u}{\|u\|_{\mathcal{W}}}$. Then we have

$\displaystyle\mathcal{J}(u)$
$\displaystyle=\frac{1}{2}\|u\|^{2}_{\mathcal{W}}-\frac{1}{2}\int_{V}u^{2}\log
u^{2}d\mu$
$\displaystyle=\frac{\|u\|^{2}_{\mathcal{W}}}{2}\left(\|\phi\|^{2}_{\mathcal{W}}-\int_{V}\phi^{2}\log\phi^{2}d\mu-\int_{V}\phi^{2}\log\|u\|^{2}_{\mathcal{W}}\right)$
$\displaystyle\leq\frac{\|u\|^{2}_{\mathcal{W}}}{2}\left(1+C_{\varepsilon}\|\phi\|^{2-\varepsilon}_{2-\varepsilon}+C_{\varepsilon}\|\phi\|^{2+\varepsilon}_{2+\varepsilon}-\|\phi\|^{2}_{2}\log\|u\|^{2}_{\mathcal{W}}\right).$

Therefore, there exists an $R=R(\widetilde{\mathcal{W}})$ such that
$\mathcal{J}\leq 0$ on $\widetilde{\mathcal{W}}\setminus
B_{R(\widetilde{\mathcal{W}})}$.

From the above argument, we know that the functional $\mathcal{J}$ satisfies
all the conditions of Lemma 3.1. Therefore, the equation (1) has a sequence of
solutions $\\{u_{n}\\}\subset\mathcal{W}$ such that
$\mathcal{J}(u_{n})\rightarrow+\infty$ as $n\rightarrow\infty$.

## Acknowledgements

This research is supported by National Natural Science Foundation of China
(No. 12101355) and the Open Project Program (K202303) of Key Laboratory of
Mathematics and Complex System, Beijing Normal University. Part of this work
was carried out while this author was visiting Tsinghua University.

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