Multiple solutions of Kazdan-Warner equation on graphs in the negative case

Shuang Liu [email protected] Yunyan Yang¹¹1Corresponding author [email protected] Department of Mathematics, Renmin University of China, Beijing 100872, P. R. China

Abstract

Let $G=(V,E)$ be a finite connected graph, and let $\kappa:V\rightarrow\mathbb{R}$ be a function such that $\int_{V}\kappa d\mu<0$ . We consider the following Kazdan-Warner equation on $G$ :

\Delta u+\kappa-K_{\lambda}e^{2u}=0,

where $K_{\lambda}=K+\lambda$ and $K:V\rightarrow\mathbb{R}$ is a non-constant function satisfying $\max_{x\in V}K(x)=0$ and $\lambda\in\mathbb{R}$ . By a variational method, we prove that there exists a $\lambda^{*}>0$ such that when $\lambda\in(-\infty,\lambda^{*}]$ the above equation has solutions, and has no solution when $\lambda\geq\lambda^{\ast}$ . In particular, it has only one solution if $\lambda\leq 0$ ; at least two distinct solutions if $0<\lambda<\lambda^{*}$ ; at least one solution if $\lambda=\lambda^{\ast}$ . This result complements earlier work of Grigor’yan-Lin-Yang [7], and is viewed as a discrete analog of that of Ding-Liu [4] and Yang-Zhu [17] on manifolds.

keywords:

Kazdan-Warnar problem on graph , variation problem on graph

MSC:

[2010] 35R02, 34B45

^†^†journal: ***

1 Introduction

Variational method is always a powerful tool in partial differential equations and geometric analysis. Recently, using this tool, Grigor’yan-Lin-Yang [7, 8, 9] obtained existence results for solutions to various partial differential equations on graphs. In particular, Kazdan-warnar equation was proposed on graphs in [7]. The Kazdan-Warner equation arises from the basic geometric problem on prescribing Gaussian curvature of Riemann surface, which systematically studied by Kazdan-Warner [12, 13]. On a closed Riemann surface $(\Sigma,g)$ with the Gaussian curvature $\kappa$ , let $\widetilde{g}=e^{2u}g$ be a smooth metric conformal to $g$ and $K$ be the Gaussian curvature with respect to $\widetilde{g}$ . Then $u$ satisfies the equation

\Delta_{g}u+\kappa-Ke^{2u}=0,

(1)

where $\Delta_{g}$ denotes the Laplace-Beltrami operator with respect to the metric $g$ . Let $v$ be a solution to $\Delta_{g}v=\overline{\kappa}-\kappa$ and $f=2(u-v)$ , where $\overline{\kappa}$ is the averaged integral of $\kappa$ . Then the above equation is transformed to

\Delta_{g}f+2\overline{\kappa}-(2Ke^{2v})e^{f}=0.

Hence, one can free (1) from the geometric situation, and just studies the equation

\Delta_{g}f+c-he^{f}=0,

(2)

where $c$ is a constant and $h$ is a function. On graphs, it seems to be out of reach to resemble this topic in terms of Gaussian curvature. Therefore, in [7], the authors focused on the equation similar to the form of (2), namely the Kazdan-Warner equation on graph, and obtained the following: when $c=0$ , it has a solution if and only if $h$ changes sign and the integral of $h$ is negative; when $c>0$ , it has a solution if and only if $h$ is positive somewhere; when $c<0$ , there is a threshold $c_{h}<0$ such that it has a solution if $c\in(c_{h},0)$ , but it has no solution for any $c<c_{h}$ . Later, Ge [5] found a solution in the critical case $c=c_{h}$ . More recently Ge-Jiang [6] studied the Kazdan-Warner equation on infinite graphs and Keller-Schwarz [11] on canonically compactifiable graphs; Camilli-Marchi [3] extended the Kazdan-Warner equation on network; for other related works, we refer the readers to [10, 14].

Let us come back to a closed Riemann surface $(\Sigma,g)$ , whose Euler characteristic is negative, or equivalently $\int_{\Sigma}\kappa dv_{g}<0$ . Replacing $K$ by $K+\lambda$ in (1) with $K\leq 0$ , $K\not\equiv 0$ , and $\lambda\in\mathbb{R}$ , Ding-Liu [4] obtained the following conclusion by using a method of upper and lower solutions and a variational method: there exists a $\lambda^{\ast}>0$ such that if $\lambda\leq 0$ , then (1) has a unique solution; if $0<\lambda<\lambda^{\ast}$ , then (1) has at least two distinct solutions; if $\lambda=\lambda^{\ast}$ , then (1) has at least one solution; if $\lambda>\lambda^{\ast}$ , then (1) has no solution. Recently, this result was partly reproved by Borer-Galimberti-Struwe [2] via a monotonicity technique due to Struwe [15, 16], and was extended to the case of conical metrics by Yang-Zhu [17].

Our aim is to extend results of Ding-Liu [4] to graphs. Let us recall some notations from graph theory. Throughout this paper, $G=(V,E)$ is assumed to be a finite connected graph. The edges on the graph are allowed to be weighted. Weights are given by a function $\omega:V\times V\rightarrow[0,\infty)$ , the edge $xy$ from $x$ to $y$ has weight $\omega_{xy}>0$ . We assume this weight function is symmetric, $\omega_{xy}=\omega_{yx}$ . Let $\mu:V\rightarrow\mathbb{R^{+}}$ be a positive measure on the vertices of the $G$ . Denote by $V^{\mathbb{R}}$ the space of real functions on $V$ . and by $\ell^{p}_{\mu}=\{f\in V^{\mathbb{R}}:\sum_{x\in V}\mu(x)|f(x)|^{p}<\infty\}$ , for any $1\leq p<\infty$ , the space of $\ell^{p}$ integrable functions on $V$ with respect to the measure $\mu$ . For $p=\infty$ , let $\ell^{\infty}=\{f\in V^{\mathbb{R}}:\sup_{x\in V}|f(x)|<\infty\}$ be the set of all bounded functions. As usual, we define the $\ell^{p}_{\mu}$ norm of $f\in\ell^{p}_{\mu},1\leq p\leq\infty$ , by

\|f\|_{p}=\left(\sum_{x\in V}\mu(x)|f(x)|^{p}\right)^{{1}/{p}},1\leq p<\infty,\,\|f\|_{\infty}=\sup_{x\in V}|f(x)|.

We define the Laplacian $\Delta:V^{\mathbb{R}}\rightarrow V^{\mathbb{R}}$ on $G$ by

\Delta f(x)=\frac{1}{\mu(x)}\sum_{y\sim x}\omega_{xy}(f(x)-f(y)).

(3)

Given the weight $\omega$ on $E$ , there are two typical choices of Laplacian as follows:

1.

$\mu(x)=\deg(x):=\sum_{y\sim x}\omega_{xy}$ for all $x\in V$ , which is called the normalized graph Laplacian;
2.

$\mu(x)\equiv 1$ for all $x\in V$ , which is the combinatorial graph Laplacian.

In this paper, we do not restrict $\mu(x)$ to the above two forms, but only require $\mu(x)>0$ for all $x\in V$ . Note that the Laplace operator defined in (3) is the negative usual Laplace operator. The gradient form is defined by

	$\displaystyle 2\Gamma(f,g)(x)$	$\displaystyle=$	$\displaystyle(f\cdot\Delta g+g\cdot\Delta f-\Delta(f\cdot g))(x)$
		$\displaystyle=$	$\displaystyle\frac{1}{\mu(x)}\sum_{y\sim x}\omega_{xy}(f(x)-f(y))(g(x)-g(y)).$

For the sake of simplicity, we write $\Gamma(f,f)=\Gamma(f)$ . Sometimes we use the notation $\nabla f\nabla g=\Gamma(f,g)$ . The length of the gradient is denoted by

|\nabla f|(x)=\sqrt{\Gamma(f)}(x).

From now on, we write $\int_{V}ud\mu=\sum_{x\in V}\mu(x)u(x)$ . Define a Sobolev space with a norm on the graph $G$ by

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

and

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

respectively. Since $G$ is a finite graph, we have that $W^{1,2}(V)$ is exactly $V^{\mathbb{R}}$ , a finite dimensional linear space. This implies the following Sobolev embedding:

Lemma 1 ([7], Lemma 5).

If $G$ is a finite graph, then the Sobolev space $W^{1,2}(V)$ is precompact. Namely, if $\{u_{j}\}$ is bounded in $W^{1,2}(V)$ , then there exists some $u\in W^{1,2}(V)$ such that up to a subsequence, $u_{j}\rightarrow u$ in $W^{1,2}(V)$ .

The Kazdan-Warner equation we are interested in this paper reads as

\Delta u+\kappa-K_{\lambda}e^{2u}=0\quad\mbox{on}\quad V,

(4)

where $\kappa\in V^{\mathbb{R}}$ is a function, and $K_{\lambda}=K+\lambda$ , $\lambda\in\mathbb{R}$ , $K\in V^{\mathbb{R}}$ is a function. Now we are ready to state our main results.

Theorem 2.

Let $G=(V,E)$ be a finite graph, $\kappa$ and $K_{\lambda}$ be given as in (4) such that $\int_{V}\kappa d\mu<0$ , $K\leq\max_{V}K=0$ , and $K\not\equiv 0$ . Then there exists a $\lambda^{*}\in(0,-\min_{V}K)$ satisfying

1.

if $\lambda\leq 0$ , then (4) has a unique solution;
2.

if $0<\lambda<\lambda^{*}$ , then (4) has at least two distinct solutions;
3.

if $\lambda=\lambda^{*}$ , then (4) has at least one solution;
4.

if $\lambda>\lambda^{*}$ , then (4) has no solution;

Remark 1.

The assertion of $\lambda^{*}<-\min_{V}K$ comes from the conclusion of Step 2 in Subsection 3.3.

Remark 2.

Compared to the existence of solutions in the literature (see for example [7, 5]), the above results firstly reveal the multiple solution problem of Kazdan-Warner equation on graphs in the negative case.

The proof of Theorem 2 is based on the method of variation. It can be viewed as a discrete analog of the result of Ding-Liu [4]. The remaining part of this paper will be organized as follows: In Section 2, we give several preliminary lemmas for our use later; In Section 3, we finish the proof of Theorem 2.

2 Preliminaries

In this section, we provide discrete versions of the maximum principle, the Palais-Smale condition and the upper and lower solution principle. Note that $G=(V,E)$ is a finite connected graph.

2.1 maximum principle

To proceed, we need the following maximum principles, which are known for experts (see for examples [7, 8]). For readers’ convenience, we include the detailed proofs here.

Lemma 3 (Weak maximum principle).

For any constant $c>0$ , if $u$ satisfies $\Delta u+cu\geq 0$ , then $u\geq 0$ on $V$ .

Proof.

Let $u^{-}=\min\{u,0\}$ . For any $x\in V$ , we claim that

\Delta u^{-}(x)+cu^{-}(x)\geq 0,

(5)

from which, one has

\int_{V}\Gamma(u^{-})d\mu+c\|u^{-}\|^{2}_{\ell^{2}_{\mu}}=\langle u^{-},\Delta u^{-}+cu^{-}\rangle\leq 0.

This leads to $u^{-}\equiv 0$ on $V$ .

To prove this claim, we first consider the case $u(x)\geq 0$ . Therefore, $cu^{-}(x)=0$ and

\Delta u^{-}(x)=\frac{1}{\mu(x)}\sum_{y\sim x}\omega_{xy}(u^{-}(x)-u^{-}(y))=-\frac{1}{\mu(x)}\sum_{y\sim x}\omega_{xy}u^{-}(y)\geq 0,

due to $u^{-}(z)\leq 0$ for any $z\in V$ . In the case $u(x)<0$ , one has $cu^{-}(x)=cu(x)$ and thus

\begin{split}\Delta u^{-}(x)=\frac{1}{\mu(x)}\sum_{y\sim x}\omega_{xy}(u^{-}(x)-u^{-}(y))&=\frac{1}{\mu(x)}\sum_{y\sim x}\omega_{xy}(u(x)-u^{-}(y))\\ &\geq\frac{1}{\mu(x)}\sum_{y\sim x}\omega_{xy}(u(x)-u(y))=\Delta u(x).\end{split}

It follows that $\Delta u^{-}(x)+cu^{-}(x)\geq\Delta u(x)+cu(x)\geq 0$ , which confirms (5) and ends the proof of the lemma. ∎

Lemma 4 (Strong maximum principle).

Suppose that $u\geq 0$ , and that $\Delta u+cu\geq 0$ for some constant $c>0$ . If there exists $x_{0}\in V$ such that $u(x_{0})=0$ , then $u\equiv 0$ on $V.$

Proof.

Let $x=x_{0}$ , we have

\frac{1}{\mu(x_{0})}\sum_{y\sim x_{0}}\omega_{yx_{0}}(u(x_{0})-u(y))+cu(x_{0})\geq 0,

which implies

\frac{1}{\mu(x_{0})}\sum_{y\sim x_{0}}\omega_{yx_{0}}u(y)\leq 0.

Since $u\geq 0$ and $\omega_{yx_{0}}>0$ for all $y\sim x_{0}$ , we obtain

u(y)=0,~{}~{}~{}~{}\mbox{for all}~{}~{}y\sim x_{0}.

Therefore, $u\equiv 0$ on $V$ by the connectedness of $G.$ ∎

2.2 Palais-Smale condition

We define a functional $E_{\lambda}:W^{1,2}(V)\rightarrow\mathbb{R}$ by

E_{\lambda}(u)=\int_{V}(|\nabla u|^{2}+2\kappa u-K_{\lambda}e^{2u})d\mu,

where $\kappa$ and $K_{\lambda}$ are given as in the assumptions of Theorem 2, in particular $\int_{V}\kappa d\mu<0$ . For any $\phi\in W^{1,2}(V)$ , denote by $dE_{\lambda}(u)(\phi)$ the Frechet derivative of the functional, by $d^{k}E_{\lambda}(u)(\phi,\cdots,\phi)$ the Frechet derivative of order $k\geq 2$ .

Lemma 5 (Palais-Smale condition).

Suppose that $V_{\lambda}^{-}=\{x\in V:K_{\lambda}(x)<0\}$ is nonempty for some $\lambda\in\mathbb{R}$ . Then $E_{\lambda}$ satisfies the $(PS)_{c}$ condition for all $c\in\mathbb{R}$ , i.e. if $(u_{j})$ is a sequence of functions in $W^{1,2}(V)$ such that $E_{\lambda}(u_{j})\rightarrow c$ and $dE_{\lambda}(u_{j})(\phi)\rightarrow 0$ for all $\phi\in W^{1,2}(V)$ as $j\rightarrow\infty$ , then there exists some $u_{0}\in W^{1,2}(V)$ satisfying $u_{j}\rightarrow u_{0}$ in $W^{1,2}(V).$

Proof.

Let $(u_{j})$ be a function sequence such that $E_{\lambda}(u_{j})\rightarrow c$ and $dE_{\lambda}(u_{j})(\phi)\rightarrow 0,$ or equivalently

\int_{V}(|\nabla u_{j}|^{2}+2\kappa u_{j}-K_{\lambda}e^{2u_{j}})d\mu=c+o_{j}(1),

(6)

\int_{V}(\nabla u_{j}\nabla\phi+\kappa\phi-K_{\lambda}e^{2u_{j}}\phi)d\mu=o_{j}(1)\|\phi\|_{W^{1,2}(V)},~{}~{}~{}~{}\forall\phi\in W^{1,2}(V),

(7)

where $o_{j}(1)\rightarrow 0$ as $j\rightarrow\infty.$

Let $\phi\equiv 1$ in (7), one has

\int_{V}(\kappa-K_{\lambda}e^{2u_{j}})d\mu=o_{j}(1)\mu(V)^{1/2},

which implies

\int_{V}K_{\lambda}e^{2u_{j}}d\mu=\int_{V}\kappa d\mu+o_{j}(1).

(8)

Inserting (8) into (6), we obtain

\int_{V}(|\nabla u_{j}|^{2}+2\kappa u_{j})d\mu=\int_{V}\kappa d\mu+c+o_{j}(1).

(9)

We now claim that $u_{j}$ is bounded in $\ell^{2}_{\mu}$ . Suppose not, there holds $\|u_{j}\|_{\ell^{2}_{\mu}}\rightarrow\infty$ . We set $v_{j}=\frac{u_{j}}{\|u_{j}\|_{\ell^{2}_{\mu}}}.$ By the Cauchy-Schwarz inequality, one has

\int_{V}\kappa\frac{u_{j}}{\|u_{j}\|^{2}_{\ell^{2}_{\mu}}}d\mu=o_{j}(1).

This together with (9) leads to

\int_{V}|\nabla v_{j}|^{2}d\mu=o_{j}(1).

(10)

Hence, $v_{j}$ is bounded in $W^{1,2}(V)$ . In view of Lemma 1 and (10), $v_{j}\rightarrow\gamma$ in $W^{1,2}(V)$ for some constant $\gamma.$ Here and in the sequel, we do not distinguish sequence and subsequence. Since $\|v_{j}\|_{\ell^{2}_{\mu}}=1$ , we have $\gamma\neq 0.$ It follows from (9) that

\int_{V}\kappa v_{j}d\mu\leq o_{j}(1).

Passing to the limit $j\rightarrow\infty$ in the above inequality, we conclude that $\gamma\geq 0$ since $\int_{V}\kappa d\mu<0$ . Therefore $\gamma>0$ .

On the other hand, for any $x\in V_{\lambda}^{-}$ , if there exists $N\in\mathbb{N}$ , if $j>N$ such that $u_{j}(x)\leq 0$ , then $\lim_{j\rightarrow\infty}v_{j}(x)\leq 0$ , which contradicts $\gamma>0$ and confirms our claim. If not, let $x_{*}\in V_{\lambda}^{-}$ , due to the finiteness of $V$ , we can choose a subsequence $\{j_{k}\}_{k=0}^{\infty}$ such that $u_{j_{k}}(x_{*})>0$ . Set

\phi(x)=\left\{\begin{array}[]{ll}u_{j_{k}}(x_{*}),&\hbox{$x=x_{*}$}\\[5.16663pt] 0,&x\not=x_{\ast}.\end{array}\right.

Then

\|\phi\|_{W^{1,2}(V)}^{2}=2\sum_{y\sim x_{*}}\omega_{x_{*}y}u_{j_{k}}^{2}(x_{*})+\mu(x_{*})u_{j_{k}}^{2}(x_{*})=(2\deg(x_{*})+\mu(x_{*}))u_{j_{k}}^{2}(x_{*}).

Substituting it into (7), we have

\Delta u_{j_{k}}(x_{*})+\kappa(x_{*})-K_{\lambda}(x_{*})e^{2u_{j_{k}}(x_{*})}\leq C^{\prime}.

(11)

Since $v_{j}\rightarrow\gamma$ , $\|u_{j}\|_{\ell^{2}_{\mu}}\rightarrow+\infty$ and $V$ has finite points, we conclude

u_{j}=(\gamma+o_{j}(1))\|u_{j}\|_{\ell^{2}_{\mu}}\quad{\rm uniformly\,\,on}\quad V.

This together with (11) leads to

\|u_{j_{k}}\|_{\ell^{2}_{\mu}}^{2}o_{j_{k}}(1)+\kappa(x_{*})-K_{\lambda}(x_{*})e^{2(\gamma+o_{j_{k}}(1))\|u_{j_{k}}\|_{\ell^{2}_{\mu}}^{2}}\leq C^{\prime},

which is impossible since $K_{\lambda}(x_{\ast})<0$ . Then our claim follows immediately.

Since $u_{j}$ is bounded in $\ell_{\mu}^{2}$ , we have $u_{j}$ is bounded in $W^{1,2}(V)$ due to the finiteness of $V$ . Therefore, by Lemma 1, there exists some $u_{0}\in W^{1,2}(V)$ such that up to subsequence, $u_{j}\rightarrow u_{0}$ in $W^{1,2}(V)$ . ∎

2.3 Upper and lower solutions principle

Let $f:V\times\mathbb{R}\rightarrow\mathbb{R}$ be a function, and $f$ is smooth with respect to the second variable. We say that $u\in V^{\mathbb{R}}$ is an upper (lower) solution to the following equation

\Delta u(x)+f(x,u(x))=0,~{}~{}x\in V,

(12)

if $u$ satisfies $\Delta u(x)+f(x,u(x))\geq(\leq)~{}0$ for any $x\in V$ . We generalize ([7], Lemma 8) to the following:

Lemma 6.

Suppose that $\varphi,\psi$ are lower and upper solution to (12) respectively with $\varphi\leq\psi$ on $V$ . Then (12) has a solution $u$ with $\varphi\leq u\leq\psi$ on $V$ .

Proof.

This is a discrete version of the argument of Kazdan-Warner ([12], Lemma 9.3), and the method of proof carries over to the setting of graphs.

Since the graph is finite, there exists a constant $A$ such that $-A\leq\varphi\leq\psi\leq A$ . One can find a sufficient large constant $c$ such that $F(x,t)=ct-f(x,t)$ is increasing with respect to $t\in[-A,A]$ for any fixed $x\in V$ . We define an operator $Lu=\Delta u+cu$ , and $L$ is a compact operator and $\mbox{Ker}(L)=\mbox{span}\{1\}$ due to the finiteness of the graph. Hence, we can define $\varphi_{j+1},\psi_{j+1}$ inductively as the unique solution to

\varphi_{0}=\varphi,~{}~{}L\varphi_{j+1}(x)=c\varphi_{j}(x)-f(x,\varphi_{j}(x)),~{}~{}\forall j\geq 0,x\in V,

\psi_{0}=\psi,~{}~{}L\psi_{j+1}(x)=c\psi_{j}(x)-f(x,\psi_{j}(x)),~{}~{}\forall j\geq 0,x\in V

respectively. Combining with the definition of upper (lower) solution and the monotonicity of $F(x,t)$ with respect to $t$ , we obtain

L\varphi_{0}(x)\leq L\varphi_{1}(x)=F(x,\varphi(x))\leq F(x,\psi(x))=L\psi_{1}(x)\leq L\psi(x),~{}~{}x\in V.

Then the weak maximum principle (see Lemma 3) yields that

\varphi\leq\varphi_{1}\leq\psi_{1}\leq\psi.

Moreover, it turns out that $\varphi_{1}$ and $\psi_{1}$ are lower and upper solution to (12) respectively. By induction, we have

\varphi\leq\varphi_{j}\leq\varphi_{j+1}\leq\psi_{j+1}\leq\psi_{j}\leq\psi,~{}~{}j=1,2,\cdots.

Since $V$ is finite, it is easy to see that up to a subsequence, $\varphi_{j}\rightarrow u_{1},\psi_{j}\rightarrow u_{2}$ uniformly on $V$ , and $u=u_{1}$ or $u_{2}$ is a solution to (12) with $\varphi\leq u\leq\psi$ on $V$ . ∎

3 Proof of Theorem 2

3.1 Unique solution in the case $\lambda\leq 0$ .

Claim 1.

$E_{\lambda}$ is strictly convex on $W^{1,2}(V)$ .

Proof.

We only need to show that there exists some constant $C>0$ such that

d^{2}E_{\lambda}(u)(h,h)\geq C\|h\|^{2}_{W^{1,2}(V)},~{}~{}\forall u,h\in W^{1,2}(V).

(13)

Suppose not, there would be a function $u\in W^{1,2}(V)$ and a function sequence $h_{j}\in W^{1,2}(V)$ such that $\|h_{j}\|_{W^{1,2}(V)}=1$ for all $j$ and $d^{2}E_{\lambda}(u)(h_{j},h_{j})\rightarrow 0$ as $j\rightarrow\infty$ . From Lemma 1, there exists $h_{\infty}\in W^{1,2}(V)$ , such that up to a subsequence, $h_{j}\rightarrow h_{\infty}$ as $j\rightarrow\infty$ in $W^{1,2}(V)$ . Since

d^{2}E_{\lambda}(u)(h_{j},h_{j})=2\int_{V}(|\nabla h_{j}|^{2}-2K_{\lambda}e^{2u}h_{j}^{2})d\mu,

and $K_{\lambda}\leq 0$ , it follows that $\int_{V}|\nabla h_{j}|^{2}d\mu\rightarrow 0$ and $\int_{V}K_{\lambda}e^{2u}h_{j}^{2}d\mu\rightarrow 0$ , which lead to $h_{\infty}\equiv c$ for some constant $c$ , and moreover

c^{2}\int_{V}K_{\lambda}e^{2u}d\mu=\lim_{j\rightarrow\infty}\int_{V}K_{\lambda}e^{2u}h_{j}^{2}d\mu=0.

It is easily seen that $\int_{V}K_{\lambda}e^{2u}d\mu<0$ by $K\not\equiv 0$ , thus $c=0$ . This contradicts

\|h_{\infty}\|_{W^{1,2}(V)}=\lim_{j\rightarrow\infty}\|h_{j}\|_{W^{1,2}(V)}=1.

Hence (13) holds. ∎

Claim 2.

For any $\varepsilon>0$ , there exist constants $C,C(\varepsilon)>0$ such that

E_{\lambda}(u)\geq(C-2\varepsilon)\|u\|_{W^{1,2}(V)}^{2}-2C(\varepsilon).

Proof.

By Young’s inequality, for any $\varepsilon>0$ , there exists a constant $C(\varepsilon)>0$ such that

\left|\int_{V}\kappa ud\mu\right|\leq\varepsilon\|u\|_{W^{1,2}(V)}^{2}+C(\varepsilon).

Thus, it is sufficient to find some constant $C>0$ such that for all $u\in W^{1,2}(V)$

\int_{V}(|\nabla u|^{2}-K_{\lambda}e^{2u})d\mu\geq C\|u\|^{2}_{W^{1,2}(V)}.

Suppose not, there would exist a sequence of functions $u_{j}$ satisfying

\int_{V}(|\nabla u_{j}|^{2}+u_{j}^{2})d\mu=1,~{}~{}\int_{V}(|\nabla u_{j}|^{2}-K_{\lambda}e^{2u_{j}})d\mu=o_{j}(1).

Clearly, $u_{j}$ is bounded in $W^{1,2}(V)$ , it follows from Lemma 1 that there exists some function $u\in W^{1,2}(V)$ such that up to a subsequence, $u_{j}\rightarrow u$ in $W^{1,2}(V)$ as $j\rightarrow\infty.$ Due to $K_{\lambda}\not\equiv 0$ and $K_{\lambda}\leq 0,$ we have

0<\int_{V}(|\nabla u|^{2}-K_{\lambda}e^{2u})d\mu=\lim_{j\rightarrow\infty}\int_{V}(|\nabla u_{j}|^{2}-K_{\lambda}e^{2u_{j}})d\mu=0,

which gets a contradiction. ∎

Proof of (1) in Theorem 2. It is a consequence of Claim 1 and Claim 2. Precisely we denote $\Lambda=\inf_{u\in W^{1,2}(V)}E_{\lambda}(u)$ . By Claim 2, we see that $\Lambda$ is a definite real number. Take a function sequence $u_{j}\in W^{1,2}(V)$ such that $E_{\lambda}(u_{j})\rightarrow\Lambda$ as $j\rightarrow\infty$ . Applying Claim 2, we have that $u_{j}$ is bounded in $W^{1,2}(V)$ . Then, in view of Lemma 1, there exists a subsequence of $u_{j}$ (still denoted by $u_{j}$ ) and a function $u_{0}\in W^{1,2}(V)$ such that $u_{j}\rightarrow u_{0}$ in $W^{1,2}(V)$ . Obviously $E_{\lambda}(u_{0})=\Lambda$ , and thus $u_{0}$ is a critical point of $E_{\lambda}$ . We also need to explain why $E_{\lambda}$ has only one critical point. For otherwise, we assume $u^{\ast}$ is another critical point of $E_{\lambda}$ . Note that $dE_{\lambda}(u_{0})=dE_{\lambda}(u^{\ast})=0$ . It follows from Claim 1, particularly from (13), that

	$\displaystyle E_{\lambda}(u_{0})$	$\displaystyle=$	$\displaystyle E_{\lambda}(u^{\ast})+dE_{\lambda}(u^{\ast})(u_{0}-u^{\ast})+\frac{1}{2}d^{2}E_{\lambda}(\xi)(u_{0}-u^{\ast},u_{0}-u^{\ast})$
		$\displaystyle\geq$	$\displaystyle E_{\lambda}(u^{\ast})+C\\|u_{0}-u^{\ast}\\|_{W^{1,2}(V)}^{2}$

for some positive constant $C$ , where $\xi$ is a function lies between $u^{\ast}$ and $u_{0}$ . Hence we have $E_{\lambda}(u_{0})>E_{\lambda}(u^{\ast})$ , contradicting the fact that $E_{\lambda}(u_{0})=\Lambda$ . This implies the uniqueness of the critical point of $E_{\lambda}$ . $\hfill\Box$

3.2 Multiplicity of solutions for $0<\lambda<\lambda^{*}$ .

Fixing $\lambda\in(0,\lambda^{*})$ , we will seek two different solutions of (4). One is a strict local minimum of the functional $E_{\lambda}$ , and the other is from mountain-pass theorem. We firstly prove the existence of $\lambda^{*}$ . Consider the case $\lambda=0$ in the equation (4) as follows

\Delta u+\kappa-Ke^{2u}=0.

(14)

For the solution $u_{0}$ of (14), the linearized equation of (14) at $u_{0}$

\Delta v-2Ke^{2u_{0}}v=0

has only a trivial solution $v\equiv 0,$ since $K\leq 0$ and $K\not\equiv 0.$ Indeed,

0\leq\int_{V}|\nabla v|^{2}d\mu=\int_{V}v\Delta vd\mu=2\int_{V}Ke^{2u_{0}}v^{2}d\mu\leq 0,

which implies $v(x)=0$ when $K(x)<0.$ In the case $K(x)=0$ , we have $\Delta v(x)=0$ . Thus

\int_{V}|\nabla v|^{2}d\mu=\int_{V}v\Delta vd\mu=0.

It follows that $v$ is a constant function and hence $v\equiv 0.$ By the implicit function theorem, there exists a small enough $s>0$ such that the equation (4) has a solution for any $\lambda\in(0,s)$ . Indeed, let $u=tv+u_{0},v\not\equiv 0$ on $V$ , we consider $G(\lambda,t)=\Delta u+\kappa-K_{\lambda}e^{u}$ . It is easy to see that $G(0,0)=0$ , $G(\lambda,t)$ and $\partial_{t}G(\lambda,t)=\Delta v-2v(K+\lambda)e^{2(u_{0}+tv)}$ are continuous on any domain $D\subset\mathbb{R}^{2}$ , Furthermore, $\partial_{t}G(0,0)=\Delta v-2Ke^{2u_{0}}v\not\equiv 0$ unless $v\equiv 0$ on $V$ . Therefore, by the implicit function theorem, there exists $s>0$ such that $t=g(\lambda)$ and $G(\lambda,g(\lambda))=0$ for any $\lambda\in(0,s)$ . In other words, $u_{\lambda}=g(\lambda)v+u_{0},\forall\lambda\in(0,s)$ is the solution of (4). Define

\lambda^{*}=\sup\{s:\mbox{the equation \eqref{eq:0} has a solution for any $\lambda\in(0,s)$}\}.

One can see that $\lambda^{*}\leq-\min_{V}K.$ For otherwise, $K_{\lambda}=K+\lambda>0$ for some $\lambda<\lambda^{*}$ . Adding up the equation (4) for all $x\in V$ , we have

0>\int_{V}\kappa d\mu=\int_{V}K_{\lambda}e^{2u}d\mu>0,

which is impossible. In conclusion, we have $0<\lambda^{*}\leq-\min_{V}K$ .

Proof of (2) in Theorem 2. We separate the proof into the following three steps.

Step 1. The existence of the upper and lower solution of (4).

Take $\lambda_{1}$ with $\lambda<\lambda_{1}<\lambda^{*}$ , let $u_{\lambda_{1}}$ be a solution of (4) at $\lambda_{1}$ . It is easily seen that $\psi=u_{\lambda_{1}}$ is a strict upper solution of (4) at $\lambda$ , namely

\Delta\psi+\kappa-K_{\lambda}e^{2\psi}>0.

Let $v$ be the solution to the following equation

\Delta v=-\kappa+\frac{1}{\mu(V)}\int_{V}\kappa d\mu.

(15)

The existence of solution to (15) was proved in [7]. Set $\varphi=v-s$ , where $s$ is a sufficiently large constant such that $\varphi<\psi$ on $V$ and

\Delta\varphi+\kappa-K_{\lambda}e^{2\varphi}=\frac{1}{\mu(V)}\int_{V}\kappa d\mu-K_{\lambda}e^{2v-2s}<0

since $\int_{V}\kappa d\mu<0$ . Therefore, $\varphi$ is a strict lower solution of (4). Let $[\varphi,\psi]$ be the order interval defined by

[\varphi,\psi]=\{u\in V^{\mathbb{R}}:\varphi\leq u\leq\psi~{}\mbox{on}~{}V\}.

The upper and lower-solution method (Lemma 6) asserts that (4) has a solution $u_{\lambda}\in[\varphi,\psi]$ on $V$ .

Step 2. $u_{\lambda}$ can be chosen as a strict local minimum of $E_{\lambda}$ .

Let $f_{\lambda}(x,t)=ct-\kappa(x)+K_{\lambda}(x)e^{2t}$ , where $c$ is sufficiently large such that $f_{\lambda}(x,t)$ is increasing in $t\in[-A,A]$ , $A$ is a constant such that $-A\leq\varphi<\psi\leq A$ on $V$ . Let $F_{\lambda}(x,u(x))=\int_{0}^{u(x)}f_{\lambda}(x,t)dt$ . It is easy to rewrite $E_{\lambda}(u)$ as

E_{\lambda}(u)=\int_{V}(|\nabla u|^{2}+cu^{2})d\mu-2\sum_{x\in V}\mu(x)F_{\lambda}(x,u(x))-\int_{V}K_{\lambda}d\mu.

It is obvious that $E_{\lambda}$ is bounded from below on $[\varphi,\psi]$ . Therefore, we denote

a:=\inf_{u\in[\varphi,\psi]}E_{\lambda}(u).

Taking a function sequence $u_{j}\subset[\varphi,\psi]$ such that $E_{\lambda}(u_{j})\rightarrow a$ as $j\rightarrow\infty.$ From it, we can get that $u_{j}$ is bounded in $W^{1,2}(V)$ , and thus up to subsequence, $u_{j}$ converges to some $u_{\lambda}$ in $W^{1,2}(V)$ and $\ell^{q}_{\mu}$ for any $q\geq 1$ , and $e^{2u_{j}}$ converges to $e^{2u_{\lambda}}$ in $\ell^{1}_{\mu}$ . Hence

E_{\lambda}(u_{\lambda})=\inf_{u\in[\varphi,\psi]}E_{\lambda}(u).

As a consequence, $u_{\lambda}$ satisfies the Euler-Lagrange equation

\Delta u_{\lambda}(x)+cu_{\lambda}(x)=f_{\lambda}(x,u_{\lambda}(x)).

From it, one can conclude that

\varphi(x)<u_{\lambda}(x)<\psi(x),\forall x\in V.

(16)

Indeed, noting that $f_{\lambda}(x,t)$ is increasing with respect to $t\in[-A,A]$ , we have

\Delta\varphi(x)+c\varphi(x)\leq f_{\lambda}(x,\varphi(x))\leq f_{\lambda}(x,\psi(x))\leq\Delta\psi(x)+c\psi(x).

One can conclude (16) by the strong maximum principle (Lemma 4), and the fact $\varphi<\psi$ . For any $h\in W^{1,2}(V)$ , we define a function $\eta(t)=E_{\lambda}(u_{\lambda}+th),t\in\mathbb{R}$ . There holds $\varphi\leq u_{\lambda}+th\leq\psi$ for sufficiently small $|t|$ . Since $u_{\lambda}$ is a minimum of $E_{\lambda}$ on $(\varphi,\psi)$ , we have $\eta^{\prime}(0)=dE_{\lambda}(u_{\lambda})(h)=0$ and $\eta^{\prime\prime}(0)=d^{2}E_{\lambda}(u_{\lambda})(h,h)\geq 0.$ Furthermore, there exists a constant $C>0$ such that

d^{2}E_{\lambda}(u_{\lambda})(h,h)\geq C\|h\|_{W^{1,2}(V)},~{}~{}~{}~{}\forall h\in W^{1,2}(V),

(17)

which implies $u_{\lambda}$ is strict local minimum of $E_{\lambda}$ on $W^{1,2}(V)$ . It remains to prove (17). We first denote

\theta:=\inf_{\|h\|_{W^{1,2}(V)}=1}d^{2}E_{\lambda}(u_{\lambda})(h,h),

which is nonnegative. It is sufficient to prove $\theta>0$ , (17) follows. Suppose $\theta=0$ , we claim that there exists some $\tilde{h}$ with $\|\tilde{h}\|_{W^{1,2}(V)}=1$ such that $d^{2}E_{\lambda}(u_{\lambda})(\tilde{h},\tilde{h})=0$ . To see this, let $h_{j}$ be a function sequence satisfying $\|h_{j}\|_{W^{1,2}(V)}=1$ for all $j$ and $d^{2}E_{\lambda}(u_{\lambda})(h_{j},h_{j})\rightarrow 0$ as $j\rightarrow\infty.$ Up to subsequence, $h_{j}\rightarrow\tilde{h}$ in $W^{1,2}(V)$ from Lemma 1, and confirms our claim. To put it another way, the functional $v\mapsto d^{2}E_{\lambda}(u_{\lambda})(v,v)$ attains its minimum at $v=\tilde{h}$ , it follows that $d^{2}E_{\lambda}(u_{\lambda})(\tilde{h},v)=0$ for all $v\in W^{1,2}(V).$ Hence, $\tilde{h}$ is a solution of the following equation

\Delta h=2K_{\lambda}e^{2u_{\lambda}}h,~{}~{}~{}~{}\lambda\in(0,\lambda^{*}).

(18)

It is easy to see that $\tilde{h}$ is not a constant. For otherwise (18) yields

0>\int_{V}\kappa d\mu=\int_{V}K_{\lambda}e^{2u_{\lambda}}d\mu=0,

which is impossible. Multiplying (18) by $\tilde{h}^{3}$ , we obtain

\begin{split}d^{4}E_{\lambda}(u_{\lambda})(\tilde{h},\tilde{h},\tilde{h},\tilde{h})&=-16\int_{V}K_{\lambda}e^{2u_{\lambda}}\tilde{h}^{4}d\mu\\ &=-8\int_{V}\tilde{h}^{3}\Delta\tilde{h}d\mu\\ &=-8\sum_{x,y\in V}\omega_{xy}(\tilde{h}^{3}(x)-\tilde{h}^{3}(y))(\tilde{h}(x)-\tilde{h}(y))\\ &=-8\sum_{x,y\in V}\omega_{xy}(\tilde{h}^{2}(x)+\tilde{h}(x)\tilde{h}(y)+\tilde{h}^{2}(y))(\tilde{h}(x)-\tilde{h}(y))^{2}<0.\end{split}

The last inequality is due to the fact $a^{2}+ab+b^{2}>0$ for any $a,b\in\mathbb{R}$ satisfying $ab\neq 0$ . Since $d^{2}E_{\lambda}(u_{\lambda}+t\tilde{h})(\tilde{h},\tilde{h})$ attains its minimum at $t=0$ , we have $d^{3}E_{\lambda}(u_{\lambda})(\tilde{h},\tilde{h},\tilde{h})=0$ , which together with $dE_{\lambda}(u_{\lambda})(\tilde{h})=0$ and $d^{2}E_{\lambda}(u_{\lambda})(\tilde{h},\tilde{h})=0$ leads to

E_{\lambda}(u_{\lambda}+\epsilon\tilde{h})=E_{\lambda}(u_{\lambda})+\frac{\epsilon^{4}}{24}d^{4}E_{\lambda}(u_{\lambda})(\tilde{h},\tilde{h},\tilde{h},\tilde{h})+0(\epsilon^{5})<E_{\lambda}(u_{\lambda})

(19)

for small $\epsilon>0.$ Let $\epsilon$ small enough such that $\varphi\leq u_{\lambda}+\epsilon\tilde{h}\leq\psi$ , thus by (19),

E_{\lambda}(u_{\lambda}+\epsilon\tilde{h})<E_{\lambda}(u_{\lambda}),

which contradicts the fact that $u_{\lambda}$ is the minimum of $E_{\lambda}$ on $[\varphi,\psi]$ . Therefore $\theta>0$ , which concludes (17).

Step 3. The second solution of (4) is given by the mountain-pass theorem.

We shall use the mountain-pass theorem due to Ambrosetti and Rabinowitz [1], which reads as follows: Let $(X,\|\cdot\|)$ be a Banach space, $J\in C^{1}(X,\mathbb{R})$ , $e_{0},e\in X$ and $r>0$ be such that $\|e-e_{0}\|>r$ and

b:=\inf_{\|u-e_{0}\|=r}J(u)>J(e_{0})\geq J(e).

If $J$ satisfies the $(PS)_{c}$ condition with $c:=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}J(\gamma(t))$ , where

\Gamma:=\{\gamma\in C([0,1],X):\gamma(0)=e_{0},\gamma(1)=e\},

then $c$ is a critical value of $J$ . In our case, $W^{1,2}(V)$ is a Banach space, and $E_{\lambda}:W^{1,2}(V)\rightarrow\mathbb{R}$ is a smooth functional.

Since $u_{\lambda}$ is a strict local minimum of $E_{\lambda}$ on $W^{1,2}(V)$ , there exists a small enough number $r>0$ such that

\inf_{\|u-u_{\lambda}\|_{W^{1,2}(V)}=r}E_{\lambda}(u)>E_{\lambda}(u_{\lambda}).

(20)

Moreover, for any $\lambda>0,$ $E_{\lambda}$ has no lower bound on $W^{1,2}(V)$ , namely, there exists $v\in W^{1,2}(V)$ such that

E_{\lambda}(v)<E_{\lambda}(u_{\lambda}),~{}~{}~{}~{}\|v-u_{\lambda}\|_{W^{1,2}(V)}>r.

(21)

To see this, we set $V_{\varepsilon}=\{x\in V:K_{\lambda}(x)>\varepsilon\}$ for small $\varepsilon>0$ . Note that $V_{\varepsilon}$ is nonempty since $\max_{V}K=0$ and $\lambda>0$ . Let $f\in W^{1,2}(V)$ be a function which equals to $1$ in $V_{\varepsilon}$ and vanishes on $V/V_{\varepsilon}$ , then

\begin{split}E_{\lambda}(tf)&=t^{2}\int_{V}|\nabla f|^{2}d\mu+t\int_{V_{\varepsilon}}\kappa d\mu-\int_{V_{\varepsilon}}K_{\lambda}e^{2t}d\mu-\int_{V/V_{\varepsilon}}K_{\lambda}d\mu\\ &\leq At^{2}+Bt+C-\varepsilon\mu(V_{\varepsilon})e^{2t}\rightarrow-\infty,~{}~{}~{}~{}t\rightarrow+\infty.\end{split}

In view of (20), (21) and Lemma 5, the mountain pass theorem of Ambrosetti and Rabinowitz gives another critical point $u^{\lambda}$ of $E_{\lambda}$ other than $u_{\lambda}$ . In particular,

E_{\lambda}(u^{\lambda})=\min_{\gamma\in\Gamma}\max_{u\in\gamma}E_{\lambda}(u),

where $\Gamma=\{\gamma\in C([0,1],W^{1,2}(V)):\gamma(0)=u_{\lambda},\gamma(1)=v\}$ , and $dE_{\lambda}(u^{\lambda})(h)=0$ for any $h\in W^{1,2}(V)$ . $\hfill\Box$

3.3 Solvability at $\lambda^{*}$ .

For any $\lambda,0<\lambda<\lambda^{*}$ , let $u_{\lambda}$ be the local minimum of $E_{\lambda}$ obtained in the previous subsection. That is, $u_{\lambda}$ is the solution of (4), and

d^{2}E_{\lambda}(u_{\lambda})(h,h)=2\int_{V}(|\nabla h|^{2}-2K_{\lambda}e^{2u_{\lambda}}h^{2})d\mu\geq 0,~{}~{}~{}~{}\forall h\in W^{1,2}(V).

(22)

Proof of (3) in Theorem 2. The crucial point in this proof is to show that $u_{\lambda}$ is uniformly bounded in $W^{1,2}(V)$ as $\lambda\rightarrow\lambda^{*}.$ If it is true, then up to subsequence, $u_{\lambda}$ converges to some $u$ in $W^{1,2}(V)$ , and $u$ is the solution of

\Delta u+\kappa-K_{\lambda^{*}}e^{2u}=0.

Hence, we aim to prove the $W^{1,2}(V)$ boundedness of $u_{\lambda}.$ To this end, we divide the proof into the following three steps.

Step 1. There exists a constant $C>0$ such that

u_{\lambda}\geq-C~{}~{}~{}~{}\mbox{on $V$}

(23)

uniformly for any $0<\lambda<\lambda^{*}$ .

Let $v$ satisfy (15), and $\varphi_{s}=v-s$ for $s>0$ . Then for sufficient large $s$ , say $s\geq s_{0}$ , $\varphi_{s}$ is a continuous family with respect to $s$ of strict lower solution of (4) at $\lambda=0$ , i.e.

\Delta\varphi_{s}+\kappa-Ke^{2\varphi_{s}}<0.

It is clear that $\varphi_{s}$ is also a strict lower solution of (4) at $\lambda\in(0,\lambda^{*})$ for any $s\in[s_{0},\infty)$ . Now, we prove that $u_{\lambda}\geq\varphi_{s_{0}}$ , and thus (23) holds. For otherwise, we can find for some $s\in(s_{0},\infty)$

u_{\lambda}\geq\varphi_{s}~{}~{}~{}\mbox{on $V$},~{}~{}~{}~{}\mbox{and}~{}~{}u_{\lambda}(\tilde{x})=\varphi_{s}(\tilde{x})~{}~{}\mbox{for some $\tilde{x}\in V$}.

It follows from the strong maximum principle (Lemma 4) that $u_{\lambda}\equiv\varphi_{s}$ , which contradicts that $\varphi_{s}$ is strict low solution of (4) at $\lambda\in[0,\lambda^{*}).$

Step 2. The set $V_{\lambda^{*}}^{-}=\{x\in V:K_{\lambda^{*}}(x)<0\}$ is not empty.

From the case (1) in Theorem 2, there exists unique solution $w_{0}$ of the equation

\Delta w+\kappa+e^{2w}=0.

Together with the solution $u_{\lambda}$ to the equation (4) at $\lambda$ , and let $v_{\lambda}=u_{\lambda}-w_{0}$ , we have

\Delta v_{\lambda}-K_{\lambda}e^{2u_{\lambda}}-e^{2w_{0}}=0.

Multiplying the above equation by $e^{-2v_{\lambda}}$ and intergrading by parts, one has

\int_{V}K_{\lambda}e^{w_{0}}d\mu=\int_{V}e^{-2v_{\lambda}}\Delta v_{\lambda}d\mu-\int_{V}e^{-2(u_{\lambda}-2w_{0})}d\mu\leq 0.

Indeed, by Green formula

\int_{V}e^{-2v_{\lambda}}\Delta v_{\lambda}d\mu=\sum_{x,y\in V}\omega_{xy}(v_{\lambda}(x)-v_{\lambda}(y))(e^{-2v_{\lambda}(x)}-e^{-2v_{\lambda}(y)})\leq 0

since $(v_{\lambda}(x)-v_{\lambda}(y))(e^{-2v_{\lambda}(x)}-e^{-2v_{\lambda}(y)})\leq 0$ for any $x,y\in V$ . Therefore

\int_{V}K_{\lambda^{*}}e^{w_{0}}d\mu=\lim_{\lambda\rightarrow\lambda^{*}}\int_{V}K_{\lambda}e^{w_{0}}d\mu\leq 0.

Suppose that $K_{\lambda^{*}}\geq 0,$ thus $K_{\lambda^{*}}\equiv 0$ . This contradicts the assumption that $K_{\lambda^{*}}$ is not a constant.

Step 3. $u_{\lambda}$ is uniformly bounded in $W^{1,2}(V)$ as $\lambda\rightarrow\lambda^{*}$ .

Fixing $x_{0}\in V_{\lambda^{*}}^{-}$ , we set $\rho\in V^{\mathbb{R}}$ to be a function which vanishes besides $x_{0}$ and $\rho(x_{0})<0$ . Consider the equation

\Delta w+\kappa-\rho e^{2w}=0,

which always has the unique solution from the case (1) in Theorem 2, set $w_{1}.$ As above, the function $v_{\lambda}=u_{\lambda}-w_{1}$ satisfies the equation

\Delta v_{\lambda}+\rho e^{2w_{1}}-K_{\lambda}e^{2(v_{\lambda}+w_{1})}=0.

(24)

Multiplying (24) by $e^{2v_{\lambda}}$ and integrating by parts over $V$ gives

\int_{V}e^{2v_{\lambda}}\Delta v_{\lambda}d\mu+\int_{V}\rho e^{2(v_{\lambda}+w_{1})}d\mu-\int_{V}K_{\lambda}e^{2(2v_{\lambda}+w_{1})}d\mu=0.

(25)

Utilizing the fact that

(e^{2a}-e^{2b})(a-b)\geq(e^{a}-e^{b})^{2},~{}~{}~{}~{}a,b\in\mathbb{R},

one can estimate the first term in (25) as follows.

\begin{split}\int_{V}e^{2v_{\lambda}}\Delta v_{\lambda}d\mu&=\frac{1}{2}\sum_{x,y\in V}\omega_{xy}(e^{2v_{\lambda}(x)}-e^{2v_{\lambda}(y)})(v_{\lambda}(x)-v_{\lambda}(y))\\ &\geq\frac{1}{2}\sum_{x,y\in V}\omega_{xy}(e^{v_{\lambda}(x)}-e^{v_{\lambda}(y)})^{2}\\ &=\int_{V}|\nabla e^{v_{\lambda}}|^{2}d\mu.\end{split}

Inserting this estimate into (25), we obtain

\int_{V}|\nabla e^{v_{\lambda}}|^{2}d\mu+\int_{V}\rho e^{2(v_{\lambda}+w_{1})}d\mu-\int_{V}K_{\lambda}e^{2(2v_{\lambda}+w_{1})}d\mu\leq 0.

(26)

On the other hand, let $h=e^{v_{\lambda}}$ in (22), yields

\int_{V}|\nabla e^{v_{\lambda}}|^{2}d\mu-2\int_{V}K_{\lambda}e^{2(2v_{\lambda}+w_{1})}d\mu\geq 0.

Together with (26), we have

\int_{V}|\nabla e^{v_{\lambda}}|^{2}d\mu\leq-2\int_{V}\rho e^{2(v_{\lambda}+w_{1})}d\mu=-2\rho(x_{0})e^{2u_{\lambda}(x_{0})}.

(27)

One may derive that $u_{\lambda}(x_{0})$ is the uniform bound, i.e. there exists a constant $C>0$ such that

e^{2u_{\lambda}(x_{0})}\leq C.

Indeed, if $u_{\lambda}(x_{0})\leq 0,$ the above inequality is obvious; if $u_{\lambda}(x_{0})>0,$ as in the proof of Lemma 5, see (11), one can get the boundedness of $u_{\lambda}(x_{0})$ as well. Together with (27), yields

\int_{V}|\nabla e^{v_{\lambda}}|^{2}d\mu\leq C^{\prime}.

(28)

Next, we claim that $e^{v_{\lambda}}$ and thus $e^{u_{\lambda}}$ is uniformly bounded in $\ell^{2}_{\mu}$ . Suppose not, we may assume that $\|e^{v_{\lambda}}\|_{\ell^{2}_{\mu}}\rightarrow\infty$ as $\lambda\rightarrow\lambda^{*}.$ Let

w_{\lambda}=\frac{e^{v_{\lambda}}}{\|e^{v_{\lambda}}\|_{\ell^{2}_{\mu}}},

then $\|w_{\lambda}\|_{\ell^{2}_{\mu}}=1$ , and $\|\nabla w_{\lambda}\|_{\ell^{2}_{\mu}}\rightarrow 0$ from (28). It follows that $w_{\lambda}$ converges to a constant in $W^{1,2}(V)$ . From (3.3), we have $w_{\lambda}(x_{0})\rightarrow 0$ , and hence $w_{\lambda}\equiv 0$ on $V$ , which contradicts $\|w_{\lambda}\|_{\ell^{2}_{\mu}}=1$ and then confirms our claim. Since $u_{\lambda}$ is bounded below by (23) in Step 1, one has the $\ell^{2}_{\mu}$ -boundedness of $u_{\lambda}$ , and thus the $W^{1,2}(V)$ -boundedness of $u_{\lambda}$ . $\hfill\Box$

3.4 No solution when $\lambda>\lambda^{*}$ .

Proof of this case is a consequence of the upper and lower solutions principle, as follows.

Proof of (4) in Theorem 2. Let $u_{\lambda_{1}}$ be the solution of (4) at some $\lambda_{1}>\lambda^{*}$ . For any $0<\lambda<\lambda_{1}$ , $u_{\lambda_{1}}$ is an upper solution of (4) at $\lambda$ . Indeed,

\Delta u_{\lambda_{1}}+\kappa-(K+\lambda)e^{2u_{\lambda_{1}}}=(\lambda_{1}-\lambda)e^{2u_{\lambda_{1}}}>0.

From (15), it is easy to get a lower bound solution $\varphi$ of (4) at $\lambda$ such that $\varphi\leq u_{\lambda_{1}}$ . By the upper and lower solutions principle, there exists a solution of (4) at $\lambda$ , which contradicts the definition of $\lambda^{*}.$ $\hfill\Box$

References

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Multiple solutions of Kazdan-Warner equation on graphs in the negative case

Abstract

keywords:

MSC:

1 Introduction

Lemma 1 ([7], Lemma 5).

Theorem 2.

Remark 1.

Remark 2.

2 Preliminaries

2.1 maximum principle

Lemma 3 (Weak maximum principle).

Proof.

Lemma 4 (Strong maximum principle).

Proof.

2.2 Palais-Smale condition

Lemma 5 (Palais-Smale condition).

Proof.

2.3 Upper and lower solutions principle

Lemma 6.

Proof.

3 Proof of Theorem 2

3.1 Unique solution in the case λ≤0\lambda\leq 0.

Claim 1.

Proof.

Claim 2.

Proof.

3.2 Multiplicity of solutions for 0<λ<λ∗0<\lambda<\lambda^{*}.

3.3 Solvability at λ∗\lambda^{*}.

3.4 No solution when λ>λ∗\lambda>\lambda^{*}.

References

3.1 Unique solution in the case $\lambda\leq 0$ .

3.2 Multiplicity of solutions for $0<\lambda<\lambda^{*}$ .

3.3 Solvability at $\lambda^{*}$ .

3.4 No solution when $\lambda>\lambda^{*}$ .