The Kazdan-Warner equation on canonically compactifiable graphs
Matthias Keller Michael Schwarz
math.AP math.FA math.MG


The Kazdan-Warner equation on graphs]
The Kazdan-Warner equation on  canonically compactifiable  graphs
M. Keller]Matthias Keller
M. Keller,   Institut für Mathematik, Universität Potsdam
	
14476  Potsdam, Germany
matthias.keller@uni-potsdam.de
M. Schwarz]Michael Schwarz
M. Schwarz,  Institut für Mathematik, Universität Potsdam
	
14476  Potsdam, Germany
mschwarz@math.uni-potsdam.de






	We study the Kazdan-Warner equation on canonically compactifiable  graphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds. 


[
    [
    May 29, 2018
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§ INTRODUCTION

In recent years various topics in the analysis on graphs received a lot of attention. While the main focus was put so far mainly on the study of linear equations such as the heat equation, the Poisson equation or fundamental topics in spectral theory, see e.g. <cit.>, there is an upcoming interest in non-linear problems. On the one  hand this concerns non-linear operators such as the p-Laplacian, <cit.>, and on the other hand non-linear equations such as the Kazdan-Warner equation <cit.>.


A major goal is often to unveil the analogies in the continuum, i.e., Riemannian manifolds, and the discrete setting, i.e. graphs. The deeper reason of these analogies stems from the fact that both the Laplacian in the continuum and in the discrete are generators of so called Dirichlet forms or from another perspective they are generators of certain Markov processes, see <cit.>. A major challenge in the discrete is often to find the correct geometric notions to even formulate analogous statements.


The Kazdan-Warner equation arises from the basic geometric question which functions are potential curvatures of 2-dimensional manifolds. Kazdan and Warner studied this question  on compact manifolds <cit.> and on open manifolds that are diffeomorphic to an open set in a compact  manifold <cit.>. From there on there is an enormous amount of work on this topic and we refer here only to <cit.> 


Compactness in the discrete setting is obviously equivalent to finiteness of the graph. For this case a very satisfying answer was found  by Grigor'yan/Lin/Yang <cit.>, see also <cit.>. On the other hand it is not a priori clear what are graph analogues to  pre-compact open manifolds. This is the starting point of our paper. We propose a class called canonically compactifiable graphs. These infinite graphs were recently introduced in <cit.>, see also <cit.>, and the analysis of Laplacians on  these graphs resembles a lot of features from the analysis of Laplacians on bounded domains or pre-compact open manifolds with sufficiently nice boundaries. For example the Dirichlet problem on the Royden boundary is uniquely solvable and the spectrum of the Laplacian is purely discrete in the case of finite measure. Here, for the Kazdan-Warner equation, we prove analogous results as in the finite case for canonically compactifiable graphs, and, indeed, our results include finite graphs as a special case.



Let us recall the original geometric motivation of the Kazdan-Warner equation.  This traces back to an equation on a 2-dimensional Riemannian manifold M with finite volume and two conformal metrics g and  g=e^2g for some smooth function . Specifically denoting the corresponding Gaussian curvatures by K and K this gives rise to the equation

    Δ_g=K-K e^2,

with the Laplace-Beltrami operator Δ_g with respect to the metric g. Provided there is a solution ψ to the equation Δ_gψ=K-K with K being the averaged curvature and letting u=2(-ψ) this translates into the equation

    Δ_g u=2K-(2Ke^2ψ)e^u.

Hence, the question whether a prescribed function K is a curvature function under a conformal transformation is equivalent to the question whether the equation above has a solution u. Of course, from the geometric picture 	 one already expects certain restrictions on K depending on K as for example that  K is positive somewhere when  K>0 or that K changes sign if K≠ 0 in the flat case K =0 (at least when M is compact).


Although there are already several very promising notions of curvatures proposed on graphs, see e.g. <cit.>, it still seems to be out of reach to study this problem in terms of one of these curvature notions. However, looking at this equation from an analytic perspective  is interesting in its own right. This reduces to the study of an equation

    Lu=-c+he^u

where L is the graph Laplacian, c is a constant, h is a function, and u is a function in the domain of L, see Section <ref> for the precise definitions.

As stated above we study this equation on so called canonically compactifiable graphs and under the assumption that the graph has finite measure. One way of characterizing canonically compactifiable graphs is that functions of finite energy are already bounded, see <cit.> (while for this work it is sufficent that all finite energy functions in ℓ^2 are bounded).  

Our results can be  summarized as follows in dependence of the sign of c and where h denotes the average of h:

	
  c=0 : There exists a solution if and only if  h<0 and h changes sign.
	
  c>0 : There exists a solution if and only if h is positive somewhere.
	
  c<0 : If there exists a solution then h<0 and if h<0 then there is -∞< c(h)<0 such that there is a solution for c(h)<c<0.

We only assume h ≠0 to be square summable. Moreover,  we consider the so-called Neumann-Laplacian which is the self-adjoint operator associated to the quadratic form on all ℓ^2-functions of finite energy. Solutions are then requested to be in the  operator domain of the Neumann-Laplacian. So, we are indeed talking about strong solutions of the equation.  From this it is clear that while the basic ideas of our proofs owe to <cit.>, they  need substantially more care on the operator theoretic level. Furthermore, in contrast to the case of manifolds we do not have a theory of elliptic regularity in the discrete case and no chain rule.



The paper is structured as follows. In the next section we introduce the setting and in particular canonically compactifiable graphs. We furthermore provide basic tools such as a Sobolev embedding theorem, a Poincar inequality and a Trudinger Moser inequality. In Section <ref> we characterize existence of solutions in the case c=0 and in Section <ref> we study the case c>0. Finally, in Section <ref> we give a necessary and sufficient criterion on the existence of solutions in the case c<0.





In this paper C denotes an arbitrary positive constant which might change from line to line. If we want to indicate that a constant C depends on a parameter β we write C=C(β).


Note added: After this work was completed we learned about the recent preprint of Ge/Jiang <cit.>. This paper studies the Kazdan-Warner equation on infinite graphs, however with different geometric assumptions on the graph and on the function h. It seems that these assumptions are independent from the assumptions of this paper and the techniques   used there are totally different from ours.



§ THE LAPLACIAN ON CANONICALLY COMPACTIFIABLE GRAPHS


In this section we introduce canonically compactifiable graphs over discrete measure spaces. These graphs, which can be seen as discrete analogues to open pre-compact manifolds,  were studied systematically in <cit.>, see also <cit.>. The definition is here slightly weakened by taking also a finite measure into consideration. We then introduce several tools that are needed for the proof. The availability of these tools such as the Sobolev embedding theorem, a Poincar inequality and a Trudinger Moser inequality, which hold trivially in the finite setting, see <cit.>,  owes here to the assumption on the graph being canonically compactifiable and of finite measure.



Let X be a countable[Here, countable means finite or countably infinite, although we are of course mainly interested in the countably infinite case] set. A symmetric function b:X× X→ [0,∞) that vanishes on the diagonal and satisfies 

    ∑_y∈ X b(x,y)<∞

for every x∈ X is called a graph on X.

A graph is called connected if for every x,y∈ X there are x_1,…, x_n∈ X such that b(x,x_1)>0, b(x_1,x_2)>0,…, b(x_n,y)>0.
We equip the set X with the discrete topology, so that every measure m on (the Borel-σ-field of)  X is given by a  function, also denoted by m, from X to [0,∞], via
m(A):=∑_x∈ A m(x). Throughout this paper we assume that  m(x)>0 for all x∈ X and that m is a finite measure, 

    m(X)<∞,

and call   (X,m) a finite discrete measure space.
Given a measure m on X we define  as the space of real valued functions on X which are p-square-summable with respect to m, 1≤ p<∞ and ℓ^∞(X) the space of bounded functions. Denote the corresponding norm ·_p by 

    f_p:=
    
    (∑_x∈ X|f(x)|^pm(x))^1/p
 Furthermore, we denote the dual pairing of ℓ^p and ℓ^q for p,q∈[1,∞] with 1/p +1/q=1 by  ⟨·,·⟩.

Denote by C(X) the space of real valued functions. We define a quadratic form :C(X)→ [0,∞] via

    (u)=1/2∑_x,y∈ X b(x,y)(u(x)-u(y))^2.

Let 
    :={u∈ C(X)|(u)<∞}.
 

Via polarization  gives rise to a semi-inner product on  also denoted by . It is easy to see that  is Markovian, i.e.

    (0∨ u∧ 1)≤(u),      u∈ .
 
By Fatou's lemma Q is lower semi-continuous and, therefore,
 the restriction  of  to 
    :=∩
 
is closed and, hence, a Dirichlet form which is often referred to as the Dirichlet form with Neumann boundary conditions or the Neumann form. We denote the form norm by

    ·_:=((·)+·_2^2)^1/2.

Furthermore, we let (f):=∞ for f∈ℓ^2(X,m)∖ D().


There is a self-adjoint operator, the Neumann Laplacian, L in correspondence with  given by 

    D()={u∈|   there is  f∈ such that 
       (u,v)=⟨ f,v⟩ holds for every  v∈},


    u=f.

From this definition it is clear that the constant functions are in D(L) in the case of finite measure.
It turns out, <cit.>,  by the virtue of Green's formula  acts as

    u(x)=1/m(x)∑_y∈ X b(x,y)(u(x)-u(y)),      u∈ D().




The goal of this paper is to study the so called Kazdan-Warner equation for functions u∈ D()

    u=-c+he^u,
 
for some given function h∈ℓ^2(X,m), h≠ 0, and a constant c.
We will study this equation on what we call canonically compactifiable graphs
over (X,m).


 A connected graph b over a finite discrete measure space (X,m) is called canonically compactifiable if 
    ⊆ℓ^∞(X).
 




	The notion of canonically compactifiable graphs b over a discrete set X (instead of a  measure space (X,m)) was introduced in <cit.> via the condition
	
    ⊆ℓ^∞(X).

	For these graphs the Royden compactification (which arises as space of maximal ideals of the commutiative Banach-algebra of  closed with respect to ·_∞) turns out to have particular nice properties. For example the Dirichlet problem with respect to the Royden boundary is uniquely solvable. Furthermore,  various results in  <cit.> suggest that canonically compactifiable stand in parallel to bounded domains in ^d. 
	



Next, we present two embedding results.


	Let b be a connected, canonically compactifiable graph over (X,m) with m(X)<∞. Then, there is a C>0 such that 
    u_∞≤ Cu_,     u∈.



	The statement follows from the closed graph theorem.	




 Let b be a connected, canonically compactifiable graph over (X,m) with m(X)<∞. Then,
  (,·_) embeds compactly into . In particular, the operator  has purely discrete spectrum.


	For the semigroup  e^-t, t≥0, of   we have by the definition of canonically compactifiable graphs and finiteness of the measure m
	
    e^-tLℓ^2(X,m)⊆ D()⊆ D()⊆ℓ^∞(X)⊆ℓ^2(X,m).
 
	Thus, e^-t  can be viewed as an operator from ℓ^2(X,m) to ℓ^∞(X) composed with the embedding ℓ^∞(X)→ℓ^2(X,m) and is, therefore, a Hilbert-Schmidt operator by the factorization principle, see <cit.>. However, compactness of e^-tL is equivalent to the compactness of the embedding of D() into ℓ^2(X,m) which is equivalent to discreteness of the spectrum of .











 


 Let b be a connected, canonically compactifiable graph over (X,m)  with m(X)<∞. Then, ()=span{1}. Furthermore,    from {1} ^⊥ to {1}^⊥ is invertible.


	The first part of the statement carries over from <cit.>. The second part now follows from the fact that by the Sobolov embedding above,  has purely discrete spectrum and is, therefore, bijective on {1}^⊥.



In the case of finite measure m we obviously have

    ℓ^∞(X)⊆ℓ^p(X,m)⊆ℓ^1(X,m),     p∈[1,∞),

and, hence,  for u∈ℓ^p(X,m) we can define

    u:=1/m(X)⟨ u,1⟩.

Next, we come to a Poincaré-inequality on canonically compactifiable graphs.


 Let b be a connected, canonically compactifiable graph over (X,m)  with m(X)<∞. Then, there is C>0 such that
 
    u-u_2^2≤ C(u),     u∈.
 


Suppose  there is a sequence (u_n) in  such that u_n=0,  (u_n)→ 0 and u_n_2=1.
Since (u_n) is bounded with respect to ·_  it contains an ℓ^2-convergent subsequence by the Sobolev embedding, Proposition <ref>. Without loss of generality we assume
 that (u_n) itself is ℓ^2-convergent to a limit  u. Obviously, u_2=1 and due to the finiteness of m, we have u =0 since
 ℓ^2-convergence induces ℓ^1-convergence (by the closed graph theorem).
 Since  is lower semicontinuous with respect to -convergence, we conclude 
    (u)≤lim inf_n→∞(u_n)=0
 and, therefore u must be constant. Thus, u=0 implies u=0 which however contradicts u_2=1. Hence the  inequality holds for every u ∈ with u=0. For general u∈ observe that u-u=0 and ( u-u) =( u).
 

Next, we discuss the following Trudinger-Moser inequality.

 Let b be a connected, canonically compactifiable graph over  (X,m) with m(X)<∞. Then, for every β∈ there is a constant C>0  such that
 
    ⟨ e^β (u-u)^2,1⟩≤ m(X)e^C|β|(u)

 for all u∈.
 In particular, for all β∈ there is C=C(β) such that 
 
    ⟨ e^β u^2,1⟩≤ C(β)

 for all u∈ with (u)=1 and u=0.


 The inequality holds for β≤ 0 since m(X)<∞. Let β>0. By Proposition <ref> and the Poincaré  inequality, Proposition <ref>, 
    (u(x)-u)^2≤u-u_∞^2≤ Cu-u_^2
     =C((u)+u-u_2)≤ C (u).

 Hence, the statement follows.  


Next, we show that for canonically compactifiable graphs with finite measure the form domain is invariant under composition with continuously differentiable functions.

 Let b be a connected, canonically compactifiable graph over (X,m) with m(X)<∞. Let u∈  and f:→ be continuously differentiable. Then, f∘ u∈.


Let u∈⊆ℓ^∞(X). Since both f and f' are continuous, we get that f∘ u,f'∘ u∈ℓ^∞(X)⊆ℓ^2(X). 
Let C:= f'|_[-u_∞,u_∞]_∞^2.
For  any
 x,y∈ X we infer 
 by the mean value theorem
 that there exists a point ξ between u(x) and u(y) such that
 
    (f∘ u(x)-f∘ u(y))^2=f'(ξ)^2(u(x)-u(y))^2<C(u(x)-u(y))^2
  
 Thus, (f∘ u)≤ C (u)<∞ and, hence, f∘ u∈.





§ THE CASE C=0





	Let b be a canonically compactifiable graph over (X,m) such that m(X)<∞ and 0≠ h∈ℓ^2(X,m). Then, there is an u∈ D() with 
    u=he^u
 if and only if h<0 and h 
	changes sign. 


 Suppose u∈ D() is a solution. Then, u∈ℓ^∞(X) since the graph is graph canonically compactifiable and, hence,    he^u∈ℓ^2(X,m). Furthermore, m(X)<∞ also implies 
 1∈ and as u is a solution, we obtain
  
    ⟨ he^u,1⟩=⟨ u,1⟩=(u,1)=0.
 
 Hence, h must change sign, as h≢0 by assumption. Then,  since e^-u∈ by Lemma <ref>, we  compute 
 
    ⟨ h,1⟩=⟨ he^u,e^-u⟩=⟨ u, e^-u⟩=(u,e^-u)<0,

 where the last inequality follows by definition of  and the fact that t↦ e^-t is monotone decreasing. Furthermore,  the inequality is strict since u can not be a constant if h changes its sign, or otherwise we would have 0=Lu=he^u and, therefore, h≡ 0 which contradicts our assumption.
 
 Let us turn to the other direction. So, assume h∈ℓ^2(X,m) is such that  h<0 and there is x_0∈ X such that 
    h(x_0)>0.
 
 Define the set 
 
    B={v∈|⟨ he^v,1⟩= ⟨ v,1⟩=0}.
 
 The strategy of the proof is to show that Q admits a minimizer on B which turns out to be a solution up to an additive constant. As a first step we show that B is non empty.

 
 
 Claim 1: The set B is not empty.
  Define v_0=1_x_0. Then, tv_0∈ for every t∈ and a simple calculation shows
 	
    ⟨ he^tv_0,1⟩=(e^t-1)h(x_0)m(x_0)+⟨ h,1⟩.
 Thus, 
    F:[0,∞)→,   t↦⟨ he^tv_0,1⟩
 is a continuous map with 
 	F(0)<0 and F(t)>0 for large t. Hence, there is t_0 such that F(t_0)=0. Then, v=t_0v_0-t_0m(x_0)/ m(X)∈ and v satisfies 
    ⟨ v,1⟩=0,
 and 
 	
    ⟨ he^v,1⟩ =
     	e^-t_0m(x_0) /m(X)⟨ he^t_0v_0,1⟩=e^-t_0 m(x_0)/m(X)F(t_0)=0.
 Thus, we get v∈ B. 
 
 

 Claim 2: There is a minimizer of  in  B.
 
 	Let (w_n) be a sequence in B such that (w_n)→inf_w∈ B(w). Using the Poincaré 
 	inequality, Proposition <ref>, we infer that (w_n) is a bounded sequence with respect to ·_. Hence, (w_n) has an -convergent subsequence by the Sobolev Embedding, Proposition <ref>, and we assume without loss of generality  that 
 	(w_n) itself is -convergent.  We denote the limit by w.
 We infer w∈ D() by  the lower semi-continuity of , i.e.,
 	
    (w)≤lim inf_n→∞(w_n)=inf_w∈ B(w).
 
 	 	 
 
 	 	Next we show w∈ B.  By Proposition <ref> we have sup_n∈w_n_∞<∞. Hence, by the finiteness of m and Lebesgue's dominated convergence theorem we observe 
    ⟨ w,1⟩=lim_n→∞⟨ w_n,1⟩=0.
 Furthermore, we have 
 	|he^w-he^w_n| ≤ |h|(e^w_∞+e^sup_n∈w_n_∞) and another application of Lebesgue's  dominated convergence theorem yields
 	
    ⟨ he^w,1⟩=lim_n→∞⟨ he^w_n,1⟩=0.

 	This shows w∈ B and, therefore, inf_w∈ B(w)≤(w). This finishes the proof of the claim.
 
 Let w be a minimizer of  on B.
 Then, the map →, f↦(f) has a minimum for f=w under the restrictions 
 
    ⟨ he^f,1⟩=0        ⟨ f,1⟩=0.
 
 We  apply the  Lagrange multiplier theorem  in Banach spaces, c.f. <cit.>.
 For this, we calculate the Frchet derivatives of the maps 
 
    Q   :D()→,    f↦(f)
    
      F   :D()→,    f↦⟨ f,1⟩
    
      H   :D()→,    f↦⟨ he^f,1⟩.

 
 Claim 3: The maps Q , F  and  H are continuously Frchet differentiable with Frchet derivatives at f∈ D()
 given by
 
    D Q[f](·)  =2(f,·),   D F[f](·)= ⟨· ,1⟩,   D H[f](·)=⟨· he^f,1⟩.

 
 	For the derivative of Q we calculate  
 	
    |(f+g)-(f)-2(f,g)|/g_   =(g)/g_≤g_^2/g_→ 0,
  
 	for g_→ 0. By the Cauchy-Schwarz-inequality the  Frchet derivative DQ[f](·) =2Q(f,·) is continuous.
 	
The statement for F follows  easily as F is linear.




 	
 	
 	Finally, for the derivative of H we estimate using that 
 	
    e^g-1-g_∞/g_≤ C∑_k=2^∞g_∞^k/k!/g_∞
     	≤ C∑_k=1^∞g_∞^k/k!=
     	 C( e^g_∞-1)→ 0
 for  g_→0	 
 	  since ·_-convergence implies ·_∞-convergence by Proposition <ref>, and 
 	 
    |⟨ he^f+g,1⟩-⟨ he^f,1⟩-⟨ hge^f,1⟩|/g_
     	 =   |⟨ he^f(e^g-1-g),1⟩|/g_
     	  	
     	  	→0

 	 	To see that the Fréchet-derivative of H is continuous, we estimate by the Cauchy-Schwarz-inequality and ·_2≤·_
 	
    |DH[f-g]()|=	|⟨φ h(e^f-e^g),1⟩|   =|⟨ h(e^f-e^g),φ⟩|
       ≤h(e^f-e^g)_2φ_

 This proves the continuity of the Frchet derivative of H.
 
 Let w be a minimizer of Q on B whose existence was proven above.
 The map 
    (DF[w],DH[w]): D()→^2,   f↦ (⟨ f,1⟩,⟨ f he^w,1⟩)
 is surjective, since there are x_0,x_1∈ X with h(x_0)>0, h(x_1)<0 by assumption, and 
 1_x_0,1_x_1∈, and, thus, DF[w](1_x_0), DF[w](1_x_1)>0 and DH[w](1_x_0)>0, DH[w](1_x_0)<0. 
 
  Thus, we can apply the Lagrange multiplier theorem in Banach spaces and infer that there are λ,μ∈ such that
 
    2(w,g)=λ⟨ ghe^w,1⟩+μ⟨ g,1⟩=
      ⟨λ he^w +μ,g⟩
 holds for every g∈, where λ he^w+μ∈ as m is finite and w∈ D() is 
 bounded, since the graph is canonically compactifiable. Therefore, by definition, w∈ D() and 
    w=λ/2 he^w+μ/2.

 
 Furthermore, by 1∈ we deduce 
    0=2(w, 1)=λ⟨ he^w,1⟩ +μ m(X).

 Since w∈ B we see ⟨ he^w,1⟩=0 and, thus, μ=0.
 
 Finally, we show λ>0. It is easy to see that λ can not be zero, since otherwise we would infer w≡ 0 and the constant zero is not in B.
 Thus, we have 0>2(w,e^-w)=λ⟨ h,1⟩, where the first inequality can be seen by an easy calculation. Since by assumption ⟨ h,1⟩=h m(X)<0, we infer λ>0. 
 Therefore, we get λ/2 =e^σ for some σ∈ and the function u=w+σ∈ D() is a solution. This concludes the proof.






§ THE CASE C>0


	Let b be a canonically compactifiable graph over (X,m) such that m(X)<∞. Let c>0 and 0≠ h∈ℓ^2(X,m). Then, there is an u∈ D() with 
    u=he^u-c
 if and only if h is positive somewhere.  


	Suppose u∈ D() is a solution. Then, by 
    0=(u,1)=⟨ u, 1 ⟩=⟨ -c,1 ⟩+⟨ he^u,1 ⟩
 we infer 
    cm(X)=⟨ he^u,1 ⟩
 and, since 
	c>0, the function h must be positive somewhere.
	
	Now suppose that h is positive somewhere. Define 
    B={v∈|⟨ he^v-c,1 ⟩ =0}
 
		and 
		
    J:D()→,    		
    		J(v):=1/2(v)+cm(X)v.

	The strategy of the proof is similar to the proof of Theorem <ref> above. We show that J assumes a minimum on B which turns out to be a solution. To this end we first show that B is non-empty and J is bounded below on B. Afterwards we show that J assumes a minimum and then we show that this minimum is a solution by the virtue of Lagrange multipliers.

	
	Claim 1: The set B is not empty. 
	
		Let x_0∈ X be such that 
		h(x_0)>0 and define v_t,l=t1_x_0-l for t,l∈. 
		We get v_t,l∈ for every t,l∈. Define F:^2→,
		
    F(t,l):=⟨ he^v_t,l,1 ⟩=(e^t-1)e^-lh(x_0)m(x_0)+e^-l⟨ h,1 ⟩

		which  is obviously continuous. Furthermore, for fixed 
		t we infer F(t,l)→ 0 for l→∞ and for fixed l we infer F(t,l)→∞ for t→∞. 
		Since the continuous image of connected sets is connected, we conclude that there are t_0,l_0∈ such that F(t_0,l_0)=cm(X) and, hence, v_t_0,l_0∈ B. 
	
Next we show that J is bounded from below on B.



	Claim 2: There is a constant C such that 
	J(v)≥1/4(v)+C, v∈ B.
	
		For v∈ B we infer
		
    e^v=cm(X)/⟨ he^v-v,1 ⟩

		from the calculation
		
    ⟨ he^v-v,1 ⟩=e^-v⟨ he^v,1 ⟩=e^-vcm(X).

		
		Therefore, 
    J(v)   =1/2(v)+cm(X)v
       =1/2(v)+cm(X)log(e^v)
       =1/2(v)+cm(X)log(cm(X))-cm(X)log(⟨ he^v-v,1 ⟩).

		The idea is now to control the third term on the right hand side by (v).
		To this end we  apply the Cauchy-Schwarz inequality, the basic inequality 
		2st≤ 2ε s^2+2/εt^2 for s=(v)^1/2, t=(v-v)/(v)^1/2 and ε>0, and, the Trudinger-Moser inequality, Proposition <ref> for the function u=(v-v)/ (v)= (v-v)/(v-v) and β=2/ε
		
    ⟨ he^v-v,1 ⟩^2≤h_2^2⟨ e^2(v-v),1 ⟩≤h_2^2⟨ e^2ε(v)+2(v-v)^2/ε(v),1 ⟩
    		≤h_2^2Ce^2ε(v).

		Therefore, by the computation for J(v) above we obtain with  ε=1/(4cm(X))
		
    J(v)
    		   ≥1/2(v)+cm(X)log(cm(X))-cm(X) log(Ch_2 e^ε(v))
    
    		   = 1/4(v)+cm(X)log(cm(X)/Ch_2)
       = 1/4(v)+C.

	
	
	Claim 3: There is a minimizer of J in B.  
	
		Let u_n be a minimizing sequence in B such that J(u_n)→inf_w∈ BJ(w)>-∞. This yields by Claim 2 and by  the Poincaré inequality, Proposition <ref>, that the sequence (u_k-u_k) is a bounded sequence in 
		(,·_). By the definition of J 
    u_k=1/m(X)(J(u_k)-(u_k))
 we infer that the sequence (u_k) is bounded in . Hence, by 
		
    u_k_≤u_k-u_k_+u_k_=u_k-u_k_+u_km(X)^1/2
 the sequence (u_k) is bounded in (,·_).
		By  the weak compactness of closed balls in Hilbert spaces (u_k) has a ·_ weakly convergent subsequence  which is without loss of generality (u_k) itself. By the Sobolev Embedding, Proposition <ref>, (u_k) converges to an u in ℓ^2(X,m) which is in D() as well.
		
		 Furthermore, by Proposition <ref> we infer
		sup_n∈u_k_∞<∞ and, hence, u is bounded as well. 	By the inequality |he^u-he^u_n| ≤ |h|(e^u_∞+e^sup_n∈u_n_∞) we can apply Lebesgue's  dominated convergence theorem to obtain
		
    ⟨ he^u,1 ⟩=lim_n→∞⟨ he^u_n,1 ⟩=cm(X).

		Therefore u∈ B and, by the lower semi-continuity of , we infer 
    J(u)≤lim inf_n→∞J(u_n)=inf_v∈ BJ(v)
 and we conclude that u is a minimizer.  
	
	Let u∈ B be a minimizer of J on B.
	Then the map J has a minimum for u∈ D() under the restriction given by the map 	I: D()→
		
    I(f):=⟨ he^f-c,1 ⟩=0.
  
	So, our next goal is to apply the theorem about Lagrange multipliers in Banach spaces, c.f. <cit.>[Theorem 43.D]. For this we 
	need to calculate the Frchet derivatives of J and I.

	
	
	Claim 4: The maps J and I  are continuously Frchet differentiable and have Frchet derivatives at f∈ D() given by
	
    DJ[f](·)=(f,·)+c⟨·,1 ⟩,     DI[f](·)=⟨· he^f,1 ⟩.

	 With the notation from the proof  of Theorem <ref> the maps J and I can be represented as
		
    J=1/2Q+cF     I= H-cF.

	Hence, the statement follows from Claim 3 of the proof of Theorem <ref>.
	  
	It is easy to see that the map DI[f](·):→ is surjective. Hence, we can apply <cit.>[Theorem 43.D] and infer the existence of λ∈ such that   
	
    (u,f)+c⟨ f,1 ⟩=λ⟨ fhe^u,1 ⟩
 holds for every f∈ and the minimizer u. 

	Hence, 
    (u,f)=⟨ -c+λ he^u,f ⟩
 for every f∈. 
	Note that λ he^u-c∈ as u is bounded and m is finite.
	Therefore, by definition, u∈ D() and 
    u=λ he^u-c.
	
	By 1∈ we infer 
    0=(u, 1)=λ⟨ he^u,1 ⟩ - cm(X).

	Since u∈ B we see ⟨ he^u,1 ⟩=cm(X) and, thus, λ=1.
	This concludes the proof.



 

§ THE CASE C<0

In this section we treat the case c<0 and prove the following theorem.


	Let b be a connected, canonically compactifiable graph over (X,m) and m(X)<∞. Let  0≠ h∈ℓ^2(X,m) and  c>0. Then, the following holds:
	 
	 	
  (1) If there is u∈ D() with u=he^u-c, then h<0.
	 	
  (2)  If h<0, then there exists a constant -∞≤ c_-(h)<0 such that the equation u=he^u-c has a solution for all 0>c>c_-(h) and no solution for every c<c_-(h). If h≤0, then c_-(h)=-∞.
	 

While the proof of (1) follows by a direct calculation, the proof of (2) involves the construction of a solution via lower and upper solutions. In particular,
we call a function u_0∈ D() a lower solution if 
    u_0≤ he^u_0-c,
  and we call a function u_1∈ D() an upper solution if 

    u_1≥ he^u_1-c.
 

We introduce for a function k:X→ (0,∞) the bilinear form 
+k   on ℓ^2(X,m) via the form sum

    D(+k)=D()∩ℓ^2(X,km)
       (+k)(f,g)=(f,g)+⟨ kf,g⟩,     f,g∈ D(+k).

An immediate consequence of Fatou's lemma (and the fact that D()=D∩ℓ^2(X,m)) yields that +k is lower semicontiuous and, hence,  closed. We denote the associated non-negative, selfadjoint operator on ℓ^2(X,m) by +k which acts as

    (+k )f(x)= f(x)+ k(x)f(x),     f∈ D(+k), x∈ X.

We show next that for canonically compactifiable graphs over  (X,m) with m(X)<∞ the domains of the forms  and +k and the corresponding operators coincide whenever k∈ℓ^2(X,m).

 
 
 
 	Let b be a  canonically compactifiable graph over (X,m), m(X)<∞ and k∈ℓ^2(X,m) with k>0. Then,
 	
    D(+k)= D().

 	Moreover, the operator L+k is bijective.
 

Let f∈. 
We estimate	by using Hlder's inequality and Proposition <ref>

    ⟨ kf,f⟩≤k_1f_∞^2≤ C k_1(f).

Since k∈ℓ^2(X,m), we infer k_1<∞ as  ℓ^2(X,m)⊆ℓ^1(X,m) in the case of finite measure. Hence, f∈ℓ^2(X,km) and, therefore,

    D()⊆ℓ^2(X,km).

We conclude

    D(+k)=D()∩ℓ^2(X,km) =D().
	

To show the equality of the operator domains let f,g∈ℓ^2(X,m). 
Let G_0=(+1)^-1 and G_k=((+k)+1)^-1. Furthermore, with slight abuse of notation we  denote the operator of multiplication by k on ℓ^2(X,m) by k.
Since 
    G_kℓ^2(X,m)⊆ D(+k)=D( )⊆ℓ^∞(X)

we have

    k G_kf∈ℓ^2(X,m)

and we calculate

    ⟨ G_0k G_kf,g⟩=⟨ k G_kf,G_0g⟩= (+k)(G_kf,G_0g)-(G_kf,G_0g)

Using the facts that (+k)(G_kf,G_0g)+⟨  G_kf,G_0g⟩=⟨ f,G_0g⟩ and  (G_kf,G_0g)+⟨  G_kf,G_0g⟩=⟨ G_kf,g⟩ we infer

    ⟨ G_0k G_kf,g⟩   =⟨ G_kf,g⟩ -⟨ f,G_0g⟩ = ⟨ (G_k-G_0)f,g⟩.

Since g was chosen arbitrarily in ℓ^2(X,m) we have for all f∈ℓ^2(X,m)

    G_kf=G_0k G_kf +G_0f

Since G_k is surjective on D(+k) and G_0ℓ^2(X,m)⊆ D() we infer the statement D(+k)⊆ D().

The  other inclusion D()⊆ D(+k) follows analogously.

Finally, we show that L+k is bijective. Note, that L+k has compact resolvent by the equality G_kf=G_0k G_kf +G_0f, f∈ℓ^2(X,m), shown above and since G_0 is compact by Proposition <ref>. It is left to show that (L+k)={0}. Let u∈(L+k). Then, we have

    0=⟨ (L+k)u,u⟩=(Q+k)(u,u)≥⟨ k u,u⟩
 and, since k> 0, we infer u≡ 0.  


Furthermore, we need the following maximum principle

	Let b be a graph over (X,m) and k∈ℓ^2(X,m), k≥0. Let u∈ D(), u not constant, such that the inequality (+k) u≤ 0 holds. Then, we get u≤ 0. 


	
	By  u_±=(± u)∨ 0∈ D(+k) one has 
    0≥⟨ ( +k)u, u_+⟩=(+k)(u,u_+)=(+k)(u_+,u_+)-(+k)(u_+,u_-).
 Hence,
	
    (+k)(u_+,u_+)≤(+k)(u_+,u_-)≤ 0,
 where the latter inequality holds by (+k)(|u|)≤ (+k)(u) as k>0 and Q is a Dirichlet form. Thus u_+≡ 0 since k>0 by assumption. 



With these preparations we  show that the existence of suitable lower and upper solutions imply the existence of a solution.


	Let b be a connected, canonically compactifiable graph over (X,m), m(X)<∞ and let c<0, 0≠h∈ℓ^2(X).
	If there is an upper solution u_1 and a lower solution u_0 such that u_1≥ u_0, then there is a solution u∈ D() with u_1≥ u≥ u_0 and u=he^u-c.


Taking the upper solution u_1∈ D() and h∈ℓ^2(X), we let
	
    k=(1∨(-h))e^u_1.

Since the graph is canonically compactifiable, and thus D()⊆ℓ^∞(X), we have e^u_1∈ℓ^∞(X) and, therefore, 

    k≥ e^inf u_1>0         k∈ℓ^2(X,m).

Hence, the operator  +k is bijective by Lemma <ref>. 
For u∈ C(X) and c from our assumption we let the right hand side of the Kazdhan-Warner equation be denoted by

    f_u:X→,     f_u=he^u-c .

If u∈ℓ^∞(X,m), in particular if u∈ D(+k)=D(), we have f_u∈. 
Starting with u_1 we inductively define

    u_j+1=(+k)^-1(f_u_j+ku_j)∈,     j≥ 1.


 We next show u_0≤…≤ u_j+1≤ u_j≤…≤ u_1 by induction. For j=2  we calculate 
 
    (+k) (u_2-u_1)= f_u_1+ku_1- u_1-ku_1= f_u_1- u_1≤ 0
 as u_1 is an upper solution.
	Using the maximum principle, Lemma <ref>, we infer u_2-u_1≤ 0. 	For j≥ 2  assume u_j≤ u_j-1≤…≤ u_1. We infer
	
    (+k) (u_j+1-u_j)   =f_u_j-f_u_j-1+ku_j-ku_j-1
       =h·(e^u_j-e^u_j-1)+k·(u_j-u_j-1)
       =(he^ξ+k)·(u_j-u_j-1)
       ≤ 0,

	where the function ξ:X→ with  u_j-1≥ξ≥ u_j is given by the mean value theorem.
	Here, the last inequality follows from the definition of k. Note that u_1≥ξ by the induction hypothesis.
	Thus, the maximum principle, Lemma <ref>, yields u_j+1≥ u_j. 
	
	Next we show u_j≥ u_0 for every j≥1.
	The case j=1 follows by the assumption  u_1≥ u_0.
	Suppose  u_j≥ u_0 for some j≥1. We calculate
	
    (+k) (u_j+1-u_0)   =f_u_j+ku_j- u_0-ku_0
       ≥ h·(e^u_j-e^u_0)+k·(u_j-u_0)
       =(he^ξ+k)·(u_j-u_0)
       ≥ 0,
 
	where the function ξ:X→ with  u_j≥ξ≥ u_0 is given by the mean value theorem.
	Thus, using the maximimum principle we infer u_0≤ u_j. 
	
	Hence, the sequence u_j is pointwise monotonically decreasing  and, thus, there is a pointwise limit u:X→ with u_0≤ u≤ u_1. Using the finiteness of m and Lebesgue's dominated
	convergence theorem, we infer u∈ and u_j-u_2→ 0, j→∞. 
	On the other hand we have u_j∈ D() by D(+k) =D(), Lemma <ref>. Hence, (+k) u_j=f_u_j-1+ku_j-1 yields
	
	
 
    u_j=f_u_j-1+ku_j-1-k u_j→ f_u
 in  by Lebesgue's  dominated
 convergence theorem.
	The closedness of the operator  yields u∈ D(). By definition of f_u=he^u-c we conclude the equality
	
    u=he^u-c.
 


By the lemma above it suffices to present a lower and an upper solution u_0 and u_1 with u_0≤ u_1 to prove Theorem <ref> (2). The next lemma shows the existence of a lower solutions in general and upper solutions for  c <0 sufficiently large.





	Let b be a connected, canonically compactifiable graph over (X,m), m(X)<∞ and let h∈ℓ^2(X,m) be such that h <0.
	
		
  (a) For all c<0, there is a lower solution  u_0∈  D().
		
  (b)  There is c_-(h) <0 such that there exists  an upper solution in u_1∈ D() for c_-(h)<c<0. If furthermore h≤ 0, then c_-(h) =-∞.
	
	Furthermore, u_0 and u_1 can be chosen such that u_0≤ u_1.
	

	Recall that the operator  is bijective on the ℓ^2 functions which are orthogonal to the constants, Proposition <ref>. We  denote this inverse with slight abuse of notation by ^-1. Then ^-1  maps the orthogonal complement of the constants into D()⊆ D()⊆ℓ^∞(X) and we
	 define
	
    v_α,β=-α^-1(h_--h_-) - β

for α,β∈ where f_± denotes the positive and negative part of a function  f, i.e., 
f_±=(± f)∨ 0.  Hence, v_α,β∈ D()⊆ℓ^∞ (X) and v_α,β→-∞ uniformly for β→∞  and α∈ fixed. We estimate -h ≤ h_- to get
	
    v_α,β-he ^v_α,β+c   ≤-α h_-+αh_-+h_-e^v_α,β+c
       =(e^v_α,β-α)h_-+αh_-+c.

	Choosing 0≤α_0≤ |c|/h_- and β _0 sufficiently large we obtain the statement (a) with u_0=v__0,β_0.
	
To show (b) let N  be such that h_N=h∨(- N ) still satisfies h_N<0. Furthermore,
 let C:=^-1(h_N-h_N) _∞  and define for
α≥0
	
    v_α=α(^-1(h_N-h_N)- C ).

	We observe that -2α C≤ v_α≤ 0. 
	We calculate 
	
    (v_α+logα)-he ^v_α+log   ≥ (v_α+logα)-h_Ne ^v_α+log
       =(α- e^v_α+logα)h_N-αh_N
       =α(1- e^v_α)h_N-αh_N.

	Next, we apply  the mean value theorem and infer that there is a function ξ:X→ [-2α C,0 ] with 1-e^v_α=-e^ξ v_α, use that 0≥ v_α≥ -2α C and denote by h_N,- the negative part of h_N to deduce

    …   =-α e^ξ v_α h_N-αh_N
       ≥α e^ξ v_α h_N,--αh_N
       ≥ -2α^2e^2α CCN-αh_N.

		Since h_N <0 there is α_0 such that
		
    C_1:=-2α_0^2e^2α_0 CCN-α_0h_N>0
 
		Hence, for 0> c_-(h)> -C_1 we infer that u_1=v__0+log_0 is an upper solution for 0>c>c_-(h). This is the first statement of (b). 
		
		Now, assume h≤0.  For
α≥0 define
	
    v_α=α(^-1(h-h)-C )

	with C:=^-1(h-h) _∞. Then, v_≤ 0 and we deduce			
	
    ( v_α+log)-he ^v_α+log= (1-e^v_)h-h≥-h

			since  α >0, v_≤0, h≤ 0.			So, for given c<0  let _c≥ 0 be such that -_ch≥ -c. Then, u_1=v__c+log_c is an upper solution.
			
			
			To see that we can choose u_0≤ u_	1 we observe that u_0,u_1∈ D()⊆ℓ^∞(X) and β_0 in the definition of u_0 can be chosen arbitrary large.



	(1): Let u be a solution. Then we have 
	
    ⟨ h,1⟩=
    	⟨
    	 he^u,e^-u⟩   =	⟨ u,e^-u⟩+c⟨ 1,e^-u⟩=(u,e^-u)+c⟨ 1,e^-u⟩<0,

	where the last inequality follows since t↦ e^-t is monotone decreasing and the definition of .
	
	(2): This follows from  Lemma <ref> and Lemma <ref>.




alpha
