Existence results for a super-Liouville equation on compact surfaces
Aleks Jevnikar Andrea Malchiodi Ruijun Wu



 We are concerned with a super-Liouville equation on compact surfaces with  genus larger than one, 
   obtaining the first non-trivial existence result for this class of problems via min-max methods.
  In particular we make use of a Nehari manifold and, 
 after showing the validity of the Palais-Smale condition,  
  we exhibit either a mountain pass or linking geometry. 



Catching a glimpse 
of the parton structure of the bound proton

    Michele Viviani
    December 30, 2023
================================================================



Keywords: super-Liouville equation, existence results, min-max methods.



2010 MSC: 58J05, 35A01, 58E05, 81Q60.




§ INTRODUCTION


The Liouville equation in two dimensions, which has the form

    -Δ u= K e^2u-K,

for some given functions K,K   on a  surface M, has been extensively studied and has wide applications in geometry and physics. 
A typical example is the prescription of curvature.
Let g be a Riemannian metric on a surface M with Gaussian curvature K=K_g and let K be a given function on M. 
The question is whether there exists a functions u ∈ C^∞(M) such that the conformal metric g=e^2ug has Gaussian curvature K, 
see e.g. <cit.>. 

Since the Gaussian curvature for g is given by e^-2u(K_g-Δ_g u), the problem is equivalent to the solvability of 
equation (<ref>). 
Observe that the conformal factor u within the conformal class of [g] can be found as a critical point of the following functional:

    I(u):=∫_M ( |∇ u|^2+2K_gu-Ke^2u) _g.


When M is a closed Riemann surface, which is the case we are interested in for this paper, 
the function K has to satisfy the Gauss–Bonnet formula with respect to the new metric g.
When K is constant with the sign compatible with the Gauss–Bonnet formula, the equation is always solvable, according to the uniformization theorem. 
For non-constant K, though not being totally solved, we have a good understanding of the problem in most cases, see e.g. <cit.> and  <cit.>. 



More recently, equation (<ref>) has been studied in the context of hyperelliptic curves and of the Painlevé equations, see <cit.> and <cit.>, respectively. 


Equation (<ref>) plays also an important role in mathematical physics. On one hand, it arises in Electroweak and Chern-Simons self-dual vortices, see <cit.>. On the other hand, it appears in the Liouville field theory with applications to  string theory, see <cit.>. See also <cit.> for a recent connection between (<ref>) and the Hawking mass.
 

Motivated by the supersymmetric extension of the Liouville theory, the authors in <cit.> 
introduced the following so-called super-Liouville functional:

    I(u,ψ):=∫_M ( |∇ u|^2+ 2K_g u-e^2u +2<(+e^u)ψ,ψ> ) _g,

where  is the Dirac operator acting on spinors ψ, see Subsection <ref> for precise definitions. In a series of works they  performed blow-up analysis and studied the compactness of the solution spaces under weak assumptions and in various setting; see e.g. <cit.> and the references therein. For the role of the super-Liouville equations in physics we refer to <cit.>. One should note that the sign conventions adopted above are adapted to the sphere case. 

In this paper we consider the problem posed on a closed Riemann surface of genus > 1. 
In this case the coefficients in the action functional need to be adapted to the Gauss–Bonnet formula. 
Let g be a Riemannian metric compatible with the given complex structure.
We are going to consider the following functional: 

    J_ρ(u,ψ)∫_M ( |∇ u|^2+2K_gu+e^2u
                +2<(-ρ e^u)ψ,ψ> ) _g,

where the parameter ρ is a positive constant. 
We are adopting a different notation from that in <cit.>, making  our choice  compatible with  equation (<ref>).
The Euler–Lagrange equation for J_ρ is 

    ELΔ_g u=   e^2u+K_g-ρ e^u|ψ|^2, 
    
    _gψ=   ρ e^uψ,

which takes the name of super-Liouville equations. The system (<ref>) clearly admits the trivial solution (u_*,0), where u_* satisfies 

    -Δ u_*= -e^2u_*-K_g

and whose existence is given by the uniformization theorem.  
This is also a ground state solution  in the sense that it has minimal critical level: this follows from the fact that  the spinorial part does not affect the critical levels, while the scalar component of the functional is coercive and convex. The latter properties 
also yield uniqueness of such a trivial solution. 
The aim of the present paper is to find a solution with non-zero spinor part, a so-called non-trivial solution. 


Conformal symmetry and reduction to uniformized case.
System (<ref>) admits a conformal symmetry in the following sense. 
Suppose that (u,ψ) is a solution of (<ref>), 
let v∈ C^∞(M) and consider the metric g e^2vg.  
There exists an isometric isomorphism β S_g→S_g of the spinor bundles corresponding to different metrics such that

    _g(e^-v/2β(ψ))
     =e^-3/2vβ(_gψ),

see e.g. <cit.>, where we are using the notation from <cit.>.
Thus the pair

    u=u-v, 
    
    ψ= e^-u/2β(ψ),

solves the system 

    Δ_gu 
     =   e^-2vΔ_g(u-v)= e^-2v(e^2u+K_g-ρ e^u|ψ|^2-Δ_g v) 
    
     =   e^2(u-v)+e^-2v(K_g-Δ_g v)-ρ e^u-v|e^-v/2β(ψ)|^2 
    
     =   e^2u+K_g
     -ρ e^u|ψ|^2, 
    _gψ
     =   ρ e^-3/2vβ(e^uψ)
         =ρ e^u-v(e^-1/2vβ(ψ))
         =ρ e^uψ,

analogous to (<ref>). 
Therefore, we can work with a convenient  background metric inside the given conformal class.
W.l.o.g., recalling that the genus is larger than one, we assume that the background metric g_0 is uniformized, meaning that K_g_0≡ -1: notice that 
such a metric is unique. 
In this case the trivial solution is given by (0,0): 
the main result of the paper is the existence of a non-trivial min-max solution obtained via a variational approach. 

 
 Let M be a closed Riemann surface of genus >1 with Riemannian metric g.
 Let g_0∈ [g] be a conformal uniformized metric, i.e. K_g_0≡ -1, and suppose that the spin structure is chosen so that 0∉(_g_0).
 Then for any ρ∉(_g_0), there exists a non-trivial solution to (<ref>).   


We stress that this is the first non-trivial existence result for this class of problems. Moreover, observe that by (<ref>)  (_g) is a conformal invariant, and the condition 0∉(_[g]) is valid for many spin structures and conformal structures, as it will be explained later. 


 Note that the spinor bundle S→ M admits global automorphisms, e.g. the quaternionic structures, which form a group.
 These are parallel with respect to ∇^s and commute with the Clifford multiplications by tangent vectors, see <cit.>.
 The functional J_ρ is thus invariant under the actions of such isometries. 
It follows that there exist more than one non-trivial solution (at least eight, which is the cardinality of  the quaternion group). 
Given a solution  (U,Ψ), an intuitive example is the antipodal solution (U,-Ψ), which is in the orbit of the quaternionic structure group actions.  


Concerning the case of genus one, i.e. when the base surface is a torus, the problem might not be well-defined. 
Indeed, if we take K and K to be zero, then the system (<ref>) has only trivial solutions of the form (a,0) where a∈. 
Meanwhile in the sphere case, where both K and K should be 1, the functional turns out to be even more strongly indefinite, and   admits 
neither the classical mountain pass nor the linking geometry. 
In the genus-one case it might be interesting to consider the case of changing-sign K̃, as it was done in 
<cit.> for the prescribed Gaussian curvature problem. 



The main difficulty in studying (<ref>)  is that the Dirac operator is strongly indefinite: the spectrum of  is real and symmetric with respect to the origin. 
The classical theory for variational problems involving Laplacians or Schrödinger operators, where the positive parts usually dominates the behavior of the functional, fails to work for Dirac type actions.
There were methods developed for general strongly indefinite variational problems, see e.g. <cit.>, but they are not directly applicable to Dirac operators.
Dirac operators usually relates more closely to the geometry and topology of the spin manifolds.
Recently several attempts have been made to attack such problems.
With suitable nonlinearities as perturbation adding to the geometric equations, T. Isobe made remarkable progress in adapting the classical theory of calculus of variations to the Dirac  setting <cit.>.    
Combined with the methods of Robinowitz-Floer homology, A. Maalaoui and V. Martino  also obtained existence results of some nonlinear Dirac type equations, see <cit.> and the references therein.
In the case of super-Liouville equations we have to deal with an exponential nonlinearity, which does not fit in the above settings. 
Moreover, we are directly facing a geometric problem without auxiliary nonlinear perturbations, which is usually harder to deal with. 


The article is organized in the following way.
In the second section we introduce some preliminaries in spin geometry and discuss existence of harmonic 
spinors depending on the genus and on the conformal class. We also introduce suitable Sobolev spaces to work 
with and the Moser-Trudinger inequality. 
In the third section we tackle the strong-indefiniteness of the functional by building a natural constraint which defines a generalized Nehari manifold N. 
We then verify the Palais–Smale condition for J_ρ|_N by showing first some a-priori bounds and then 
proving strong subsequential convergence. 
For suitable ρ we finally show either mountain pass or linking geometry on the Nehari manifold which yield
the existence of a min-max critical point for J_ρ: the 
details of this construction are given in the last section. 


Acknowledgments.
A.M. has been partially supported by the projects Geometric Variational Problems and Finanziamento a supporto della ricerca di base from Scuola Normale Superiore. 
A.J. and A.M. has been partially supported by MIUR Bando PRIN 2015 2015KB9WPT_001. They are also members of GNAMPA as part of INdAM.
A.J. and R.W. are supported by the  Centro  di  Ricerca  Matematica  ‘Ennio  de  Giorgi’. 






§ PRELIMINARIES


We will assume some background in spin geometry and Sobolev spaces.
For detailed material one can refer to <cit.>. 




 §.§ Spinor bundles and Dirac operator
 
Here we introduce our setting and fix the notation.  
Let M be a closed Riemann surface with a fixed conformal structure and of genus .  
Let g be a Riemannian metric in the given conformal class and denote the Gaussian curvature by K_g. 
The orthonormal frame bundle P_(M,g)→ M is then a principal (2) bundle.
Let (2)= U(1)→(2) be the two-fold covering of the circle. 
A spin structure is given by a principal (2) bundle P_(M,g)→ M together with an equivariant two-fold covering

    P_(M,g)→ P_(M,g).

In dimension two such double coverings always exist; moreover they are in one-to-one correspondence with the elements in H^1(M;ℤ_2), see e.g. <cit.>.
This cohomology group has cardinality 2^2. 

Let S≡ S_g→ M be the associated spinor bundle with a real Riemannian structure g^s and induced spin connection ∇^s:
sections of S are called spinors.
Recall that the Dirac operator  acting on spinors is defined as the composition of the following chain

    Γ(S)Γ(T^*M⊗ S)Γ(TM⊗ S)Γ(S),

where the second isometric isomorphism is given by the identification via the metric g,  the third arrow  denotes the Clifford multiplication, and the (S)-valued map TM→(S) satisfies the following Clifford relation:

    (X)(Y)+(Y)(X)=-2g(X,Y),     ∀ X,Y∈Γ(TM).

Later, for simplicity, we will write X·ψ for (X)ψ, where X∈Γ(TM) and ψ∈Γ(S).
In terms of a local orthonormal frame (e_i)_i=1,2 we then have the Dirac operator

    ψ=∑_i(e_i)∇^s_e_iψ,  ∀ψ∈Γ(S).

This is a self-adjoint elliptic operator of first order, and it has a finite-dimensional kernel consisting of harmonic spinors.
The dimension of the space of harmonic spinors is a conformal invariant, but it depends on the choice of spin structures and the conformal structures in general.
The Bochner-Lichnerowicz formula

    ^2=(∇^s)^* ∇^s+Scal/4

implies that there is no non-trivial harmonic spinor if Scal≥0 and Scal≢0. 
In particular, there is no harmonic spinor on the 2-sphere with arbitrary metric (since there is only one conformal structure on the 2-sphere).
However, when the genus  is greater than or equal to 1, there might exist non-trivial harmonic spinors for some choice of spin structures. 
The dimensions of the spaces of harmonic spinors have been computed in literature e.g. <cit.>. 
We summarize some facts here to have a picture of the different cases.




 §.§ Examples of Riemann surfaces with no non-trivial harmonic spinors

Here we give some examples of Riemann surfaces having negative Euler characteristic 2πχ(M)=4π(1-)<0 but admitting no non-trivial harmonic spinors. 

Any element α∈ H^1(M,ℤ_2) determines a spin structure ξ(α), as well as a holomorphic line bundle ℒ_α such that ℒ_α⊗_ℒ_α= 𝒦_M, where 𝒦_M denotes the canonical line bundle of M, see e.g. <cit.>.
Denote by 𝒪(ℒ_α) the sheaf of germs of holomorphic sections of the holomorphic line bundle ℒ_α, and set h_α,g^0 = H^0(M, 𝒪(ℒ_α)). 
If the associated spinor bundle S≡ S(α,g) admits a space of harmonic spinors of dimension h_ξ(α),g, then

    h_ξ(α),g=2h^0_α,g.

It is known that, for a Riemann surface M of genus , there are precisely 2^-1(2^+1) spin structures α on M for which  h^0_α,g is an even number (such spin structures are called even spin structures on M), and for the other 2^-1(2^-1) spin structures the number h^0_α,g is odd (odd spin structures). 
 
For =1  M is topologically a torus, and for any conformal structure [g] we have four spin structures: three even spin structures with no non-trivial harmonic spinors and one odd spin structure (the trivial one α= 0) with one-dimensional space of positive harmonic spinors (hence h_ξ(0);g=2). 

For =2 the description is similar, namely for any conformal structure [g] there are ten even spin structures with no non-trivial harmonic spinors and six odd spin structures with one-dimensional space of positive harmonic spinors (hence h_ξ(0)=2).

These are the known cases where the dimension of (D) is independent of the choice of metric g (i.e. the choice of the Riemann surface structure on M). 
When the genera become larger, the dimension of the kernels generally depends on the conformal class.
Even in this case we still have  many examples where there are no non-trivial harmonic spinors.

Recall that a hyperelliptic Riemann surface is a complex projective curve admitting a rational surjective map onto P^1 which is 2-to-1 up to a finite set of branching points.
All Riemann surfaces of genera ≤ 2 are hyperelliptic, while there exist non-hyperelliptic surfaces of all genera ≥ 3. 

For the hyperelliptic case, C. Bär <cit.> showed that the spin structures correspond one-to-one to the pairwise inequivalent square roots of the canonical divisor, and in terms of a suitably defined weight of the divisors, he also clarified the dimensions h^0 of the kernels: 

 
  * if M is hyperelliptic with =2k+1, 
       
        
  * there is exactly one spin structure of weight -1 and in this case h^0=+1/2=k+1;
        
  * for w=1,3,5,⋯,-2, there are exactly 2+2-w spin structures of weight w and in this case h^0=w+1/2;
        
  * there are exactly 2+1 spin structures of weight -1 and in this case h^0=0;
       
  
  * if M is hyperelliptic with =2k,
        
         
  * there is exactly 2+2 spin structure of weight -1 and in this case h^0=[+1/2]=k;
        
  * for w=1,3,5,⋯,-1, there are exactly 2+2-w spin structures of weight w and in this case h^0=w+1/2;
        
  * there are exactly 2+1 spin structures of weight -1 and in this case h^0=0;
        


For  non-hyperelliptic surfaces, there are also known examples where the dimensions of kernels are computed. 

For a genus =3 non-hyperelliptic surface, among the 2^2=64 spin structures there are 28 odd ones with h^0=1 and 36 even ones with h^0=0. 

The case for =4 non-hyperelliptic surfaces is different: there are in total 2^2= 256 spin structures, 120 of them are odd with h^0=1, and for the other 136 even spin structures, one of the followings may happen:

 
  (I) there exists a unique even spin structure with h^0=2, while the other 135 even spin structures have h^0=0;
 
  (II) all the 136 spin structures have h^0=0.

A non-hyperelliptic Riemann surface is called of type (I) or (II) if  it satisfies the corresponding above conditions. 
Both classes are non-empty. 




 §.§ Sobolev spaces for spinors


The spinor bundle S=S_g has a Riemannian structure g^s and a spin connection ∇^s induced from the Levi-Civita connection.
Then we can define the usual Sobolev spaces with integer differentiability, namely  W^k,p(S) consists of the spinors whose k-th covariant derivatives are in L^p for k∈ℕ and p∈ [1,+∞] and W^-k,q(S) (W^k,p(S))^* where q is the Hölder conjugate of p. 
Here we will consider also fractional Sobolev exponents, see the discussion in the sequel.  

Recall that =_g is a first order elliptic operator which is essentially self-adjoint. 
The spectrum () is discrete and consists of real eigenvalues, _0()∪{λ_k}_k∈ℤ\{0}, where _0() stands for the  zero element in the spectrum (or the empty set) while the lambda's are the non-zero eigenvalues, indexed by ℤ_*≡ℤ\{0} in an increasing order (in absolute value) and counted with multiplicities:

    -∞←⋯≤λ_-l-1≤λ_-l≤⋯≤λ_-1≤ 0 
     ≤λ_1≤⋯≤λ_k ≤λ_k+1≤⋯→ +∞.

Moreover, the spectrum is symmetric with respect the the origin when M=2. 
Let φ_k be the eigenspinor corresponding to λ_k,  k∈ℤ_* with φ_k_L^2(M)=1, and let  φ__0,j,  1≤ j≤ h^0, be an orthonormal basis of (). 
These together form a complete orthonormal basis of L^2(S): 
 any spinor ψ∈Γ(S) can be  expressed in terms of this basis as 

    ψ=∑_k∈ℤ_*a_kφ_k
       +∑_1≤ j≤ h^0 a_0,jφ__0,j,

and the Dirac operator acts as

    ψ= ∑_k∈ℤ_*λ_k a_k φ_k.

For any s>0, the operator ||^sΓ(S)→Γ(S) is defined as 

    ||^sψ =∑_k∈ℤ_* |λ_k|^s a_kφ_k,

provided that the right-hand side belongs to L^2(S). The domain of ||^s is 

    H^s(S){ψ∈ L^2(S)|∫_M<||^sψ,||^sψ>_g <∞},

which is a Hilbert space with inner product

    <ψ,ϕ>_H^s 
     = <ψ,ϕ>_L^2
     +<||^sψ, ||^sϕ>_L^2.

For s=k∈ℕ, H^k(S)=W^k,2(S) and the above norm is equivalent to the Sobolev W^k,2 norm.
For s<0, H^s(S) is by definition the dual space of H^-s(S). 

Since S has finite rank, the general theory for Sobolev's embedding on closed manifold continues to hold here.
In particular, for 0<s<1 and q≤2/1-s, we have the continuous embeddings 

    H^s(S) ↪ L^q(S).

Furthermore, for q<2/1-s the embedding is compact, 
 see e.g. <cit.> for more details. 

We will mainly be interested in the case s=1/2, for which 2/1-s=4.
This is the largest space on which the Dirac action of the form

    ψ↦∫_M <ψ,ψ>_g

is well-defined. 
Note that for ψ∈ H^1/2(S) we have ψ∈ H^-1/2(S), which is defined in the distributional sense. 
Thus we can define the duality pairing

    <ψ,ψ>_H^-1/2× H^1/2∈.

On the other hand, by the expression (<ref>) we see that the function

    g^s_x (ψ(x),ψ(x))

is in L^1(M), whose integral is exactly given by ∑_k∈ℤ_*λ_k a_k^2 <∞.
By this we validate the Dirac action in the equivalent form

    <ψ,ψ>_H^-1/2× H^1/2
     =∫_M <ψ(x),ψ(x)>_g^s(x)_g(x).



Suppose h^0=0, i.e. there are no non-trivial harmonic spinors.
Then the Dirac operator  is invertible.  
Splitting into the positive and negative parts of the spectrum (), we have the decomposition 

    H^1/2(S)= H^1/2,+(S)⊕ H^1/2,-(S).

Let ψ=ψ^+ +ψ^- be decomposed accordingly: then, 

    ∫_M<ψ^+,ψ^+>_g 
     =∫_M<||^1/2ψ^+,||^1/2ψ^+>_g ≥λ_1(_g)ψ^+_L^2(M)^2,

where λ_1 is the first positive eigenvalue of =_g.
Hence

    ψ^+^2_H^1/2
     =   ψ^+^2_L^2+||^1/2ψ^+_L^2^2 
    ≤   (λ_1(_g)+1)||^1/2ψ^+^2_L^2≤(λ_1(_g)+1)ψ^+^2_H^1/2.

That is, for a given g, the integral ∫_M <ψ^+,ψ^+>_g defines a norm on H^1/2,+(S) equivalent to the Hilbert's. 
Similarly, on H^1/2,-(S) there is an equivalent norm given by

    -∫_M <ψ^-,ψ^->_g
     =||^1/2ψ^-^2_L^2.

Consequently,

    ∫_M [ <ψ^+,ψ^+>-<ψ^-,ψ^-> ] _g
     =||^1/2ψ^+^2_L^2
     +||^1/2ψ^-^2_L^2

defines a norm equivalent to the H^1/2-norm. 




 §.§ Moser–Trudinger embedding

Another space we use frequently is H^1(M)=W^1,2(M,). 
Consider the subspace in H^1(M) of the functions with zero average 

    H^1_0(M){u∈ H^1(M)|∫_M u_g=0}.

Then H^1(M)= ⊕ H^1_0(M), and any u ∈ H^1(M) can be written as u=u̅+u where u̅=_M u_g denotes the average of u. By Poincaré's inequality, ∇u_L^2 defines a norm equivalent to u_H^1 on H^1_0(M), and 

    |u̅|+∇u_L^2

 a norm equivalent to u_H^1. 
The Sobolev embedding theorems imply that for any p<∞,  H^1(M) embeds into L^p(M) continuously and compactly. 
Furthermore, the Moser–Trudinger inequality states that there exists C>0 such that 

    ∫_M exp(4π|u|^2/∇u^2_L^2(M))_g≤ C.

As a consequence 

    8πlog∫_M e^u_g
     ≤1/2∫_M |∇u|^2_g+ C.

This implies that e^u is L^p integrable for any p> 0. 
Moreover, the map

    H^1(M)∋ u↦ e^u∈ L^1(M)

is compact (see e.g. <cit.>). 
It follows that the maps H^1(M)∋ u↦ e^u∈ L^p(M) are compact for all p > 0.  




§ A NATURAL CONSTRAINT AND THE PALAIS-SMALE CONDITION


It is standard to prove that the functional

    J_ρ H^1(M)× H^1/2(S)→

defined in formula (<ref>)  is  of class C^1. 
The critical points of J_ρ, which are weak solutions of (<ref>), are actually smooth.
To see this we can use the argument from <cit.>.
Note that, although the authors there are using different Banach spaces, the proof goes quite similarly and is omitted here.
Alternatively, note that u∈ H^1(M) implies e^u∈ L^p(M) for any p<∞,i.e. the equation is actually subcritical and we can appeal to a bootstrap argument to obtain the full regularity. 
 
To obtain a non-trivial solution to the system (<ref>) we employ a min-max approach.
As observed, thanks to the conformal covariance of the system, it is sufficient to consider the uniformized metric. 
From now on we assume that g has constant Gaussian curvature K≡-1. 
For this choice we then look for non-trivial critical points of the functional

    J_ρ(u,ψ)
     =∫_M ( |∇ u|^2-2u+e^2u+2(<ψ,ψ>-ρ e^u|ψ|^2) )_g,

which are non-trivial solutions of the system

    EL_0Δ_g u=e^2u-1-ρ e^u|ψ|^2, 
    
    _gψ= ρ e^uψ.

The argument in the sequel is simplified by this assumption, but it can be modified and adapted to a general metric. 
Note that in the uniformized case the Gauss-Bonnet formula yields

    vol(M,g)=-2πχ(M)=4π(-1).


Observe that in the functional J_ρ the first part is coercive and convex. The main difficulty is due to the spinorial part which is strongly indefinite. 
To overcome this issue we are inspired by an idea from <cit.> and we consider a natural constraint: in the next section we will find critical points of the restricted functional. 




 §.§ A Nehari type manifold

Roughly speaking, the space H^1/2,-(S) defined in (<ref>) contains infinitely-many directions decreasing the functional J_ρ to negative infinity and the usual variational approaches can not be applied. 
Hence we introduce a natural constraint in order to exclude most of these directions, obtaining a submanifold in H^1(M)× H^1/2(S), which we still call it a Nehari manifold, though it may not fit into the classical definition as in <cit.>.
This may be considered to be a Nehari manifold in the generalized sense, as in <cit.>.  

Let P^± H^1/2(S)→ H^1/2,±(S) be the orthonormal projection according to the splitting in (<ref>).
Consider the map

    G H^1(M)× H^1/2(S)   → H^1/2,-(S), 
    
     (u,ψ)   ↦ P^-[(1+||)^-1(ψ-ρ e^uψ)].

Some explanations are in order. 
Recall that H^1/2,- is a Hilbert space, with inner product

    <ψ,φ>__H^1/2
     =   <ψ,φ>_L^2
     +<||^1/2ψ,||^1/2φ>_L^2
    
     =   <(1+||)ψ,φ>_H^-1/2× H^1/2.

Now, let  G(u,ψ) be the element in H^1/2(S) such that, for any φ∈ H^1/2(S), 

    <G(u,ψ),φ>__H^1/2
     =<ψ-ρ e^uψ,P^- (φ)>_H^-1/2× H^1/2.

It follows that G(u,ψ) ∈ H^1/2,-(S), and it is given by the Riesz representation theorem as 

    G(u,ψ)=P^-[(1+||)^-1(ψ-ρ e^uψ)].


Define the Nehari manifold  N=G^-1(0), which is non-empty since (u,0)∈ N for any u∈ H^1(M). 
Note that, for each u fixed, the subset 

    N_u{ψ∈ H^1/2(S)| (u,ψ)∈ N}=[ P^-∘(1+||)^-1∘ (- ρ e^u) ]

is a  linear subspace (of infinite dimension).
Hence we have a fibration N→ H^1(M)  with fiber N_u over u∈ H^1(M).
The total space N is contractible. 

 The Nehari manifold N is a natural constraint  for J_ρ, namely every critical point of J_ρ|_N is 
 an unconstrained critical point of J_ρ.  


 To see that N is a manifold, we show for any (u,ψ) the surjectivity of the differential G(u,ψ), which is given by
 
    G(u,ψ)[v,ϕ]=P^-[(1+||)^-1(ϕ-ρ e^uϕ-ρ e^u vψ)].

 Restricting to those vectors with v=0 and ϕ∈ H^1/2,-(S), we have 
 
    < G(u,ψ)[0,ϕ],ϕ>_H^1/2
      =   <(1+||)^-1(ϕ-ρ e^uϕ),ϕ>_H^1/2
    
      =   ∫_M <ϕ-ρ e^uϕ,ϕ>_g 
    
      =   -||^1/2ϕ_L^2^2-ρ∫_M e^u|ϕ|^2_g.
 
 Thus < G(u,ψ)[0,ϕ],ϕ>_H^1/2 yields a negative-definite quadratic form on H^1/2,-(S).
 In particular, G(u,ψ) is surjective onto H^1/2,-(S), for any (u,ψ). 
 It follows from the regular value theorem (for an infinite dimensional version, see e.g. <cit.>) that N=G^-1(0) is a submanifold of H^1(M)× H^1/2(S).  
 
 Next, we need to show that if (u_0,ψ_0) is a critical point of J_ρ|_N, then it is also a critical points of J_ρ on the full space H^1(M)× H^1/2(S). 

 Recall that the orthonormal basis (φ_k) for H^1/2(S) consists of eigenspinors. 
 Note that
 
    (u,ψ)∈ N
         ⇔  G(u,ψ)=0 
      ⇔∫_M <ψ-ρ e^u ψ, h>_g=0,   ∀ h ∈ H^1/2,-
       ⇔ G_j(u,ψ)∫_M <ψ-ρ e^uψ,φ_j>_g=0,   ∀ j<0,

 that is
 
    N= G^-1(0)=⋂_j<0 G_j^-1(0).

 Now let (u_0,ψ_0) be a critical point of J_ρ|_N:  ∇^N J(u_0,ψ_0)=0.  
 Then there exist μ_j∈ such that[To see that such an infinite dimensional version of the Lagrange multiplier theory works, we note that
 
    ∇^N J(u_0,ψ_0)= ∇ J(u_0,ψ_0)-(∇ J(u_0,ψ_0))^

 where ∇ J(u_0,ψ_0) denotes the unconstrained gradient and (∇ J(u_0,ψ_0))^ denotes its normal component.
 Since the gradients {∇ G_j(u_0,ψ_0): j<0} span the normal space, we can express (∇ J(u_0,ψ_0))^ in terms of them:
 
    (∇ J(u_0,ψ_0))^=∑_j<0μ_j∇ G_j(u_0,ψ_0) ∈ H^1/2(S)

 for some μ_j∈,  j<0. 
 ]
 
    J_ρ(u_0,ψ_0)=∑_j<0μ_j  G_j(u_0,ψ_0).

 Testing both sides with tangent vectors of the form (0,h), we have
 
    ∫_M<ψ_0-ρ e^u_0ψ_0,h>_g
      =∑_j<0μ_j∫_M< h-ρ e^u_0h, φ_j>_g.

 In particular, take h = φ=∑_j<0μ_jφ_j∈ H^1/2,- to obtain
 
    0=∫_M<ψ_0-ρ e^u_0ψ_0,φ>
      =∫_M<φ-ρ e^u_0φ,φ>_g
      ≤ -Cφ^2-∫_M ρ e^u_0|φ|^2_g.

 Thus φ=0, i.e.  μ_j=0 for all j<0.  
 Hence in (<ref>) we have J_ρ(u_0,ψ_0)=0. 






 §.§ Verification of the Palais-Smale condition

This subsection is devoted to verifying the (PS) condition for the constrained functional J_ρ|_N. 
Note that

    J_ρ(u,ψ)[v,ϕ]
     =∫_M 2(-Δ u-1+e^2u-ρ e^u|ψ|^2)v + 4<ψ-ρ e^uψ,ϕ>_g

and for each j<0, with G_j defined as in the above proof: 

    G_j(u,ψ)[v,ϕ]
     =∫_M <ϕ-ρ e^uϕ,φ_j>_g -∫_M ρ e^u v<ψ,φ_j>_g.

For each (u,ψ)∈ N, there exist constants μ_j(u,ψ) such that 

    ^N J_ρ(u,ψ)
     = J_ρ(u,ψ)-∑_j<0μ_j(u,ψ) G_j(u,ψ),

that is such that for any (v,ϕ)∈ H^1(M)× H^1/2(S)

    ^N J_ρ(u,ψ)[v,ϕ]
     = J(u,ψ)[v,ϕ]-∑_j<0μ_j(u,ψ) G_j(u,ψ)[v,ϕ].

Formally  writing φ(u,ψ)∑_j<0μ_jφ_j, then 

    ^N J(u,ψ)[v,ϕ]
    =   ∫_M 2(-Δ u-1+e^2u-ρ e^u|ψ|^2+ρ e^u<ψ, φ(u,ψ)>)v_g  
       + ∫_M 4 ( <ψ-ρ e^uψ,ϕ>-<φ-ρ e^uφ,ϕ> ) _g.

Note that this holds for arbitrary (v,ϕ), not only those tangent vectors to N.

Now let (u_n,ψ_n)∈ N be a (PS)_c sequence for J_ρ|_N: this will satisfy 

    J_ρ(u_n,ψ_n)=∫_M [ |∇ u_n|^2-2u_n+e^2u_n+2(<ψ_n,ψ_n>-ρ e^u_n|ψ_n|^2) ] _g→ c,


    P^-∘(1+||)^-1∘ (ψ_n-ρ e^u_nψ_n)=0;

moreover, since the differential of J_ρ  is tending to zero only when applied 
to vectors tangent  to N, there exists 
 some φ_n∈ H^1/2,-(S) 
such that 

    2(-Δ u_n-1+e^2u_n-ρ e^u_n|ψ_n|^2)-ρ e^u_n<ψ_n,φ_n>=α_n→ 0  in   H^-1(M),


    4(ψ_n-ρ e^u_nψ_n)-(φ_n-ρ e^u_nφ_n)=β_n → 0  in  H^-1/2(S).







With the same notation as above, we have 

 
  * The auxiliary spinors φ_n satisfy φ_n_H^1/2→ 0 as n→∞. 
 
  * The sequence (u_n,ψ_n) is uniformly bounded (with bounds depending on the level c) in H^1(M)× H^1/2(S).
 



 (1) Testing (<ref>) against φ_n we find 
     
    ∫_M <ψ_n-ρ e^u_nψ_n,φ_n>_g=0,

     while testing  (<ref>) against φ_n we get 
	
    -∫_M <φ_n,φ_n>_g +ρ∫_M e^u_n|φ_n|^2_g=<β_n,φ_n>.
	     
    Since φ_n lies in the span of the negative eigenspinors, we see that 
    
    Cφ_n_H^1/2^2+ρ∫_M e^u_n|φ_n|^2_g=o(φ_n_H^1/2).

    It follows that as n→∞, 
    
    φ_n_H^1/2→ 0,      ∫_M ρ e^u_n|φ_n|^2_g → 0.
    
    


(2) Testing (<ref>) against v≡ 1∈ H^1(M), we obtain 
	
    2∫_M e^2u_n_g -2∫_M _g-2ρ∫_M ( e^u_n|ψ_n|^2-e^u_n<ψ_n,φ_n> )_g
    	=<α_n,1>_H^-1× H^1,

	which can be read as 
	
    ∫_M e^2u_n_g
    	=4π(-1)+ρ∫_M e^u_n|ψ_n|^2_g+1/2ρ∫_M e^u_n<ψ_n,φ_n>_g +o(1).

	Now we can control the second integral on the right-hand side by
	
    |ρ/2∫_M e^u_n<ψ_n,φ_n>_g|
    	≤ε∫_M ρ e^u_n|ψ_n|^2_g+ε∫_M e^2u_n_g+ C(ε,ρ)φ_n^4,

    where ε>0 is some small number. 
    Substituting this into (<ref>) and noting that φ_n=o(1), we get
    
    ∫_M e^2u_n≥4π(-1)/1+ε+1-ε/1+ε∫_M ρ e^u_n|ψ_n|^2_g+o(1).

    
	Testing  (<ref>) against ψ_n we deduce 
	
    4∫_M ( <ψ_n,ψ_n>-ρ e^u_n|ψ_n|^2 )_g -∫_M <φ_n-ρ e^u_nφ_n,ψ_n>_g
    	=<β_n,ψ_n>_H^-1/2× H^1/2.

	Since the second integral vanishes because of (<ref>), we thus get
	
    ∫_M ( <ψ_n,ψ_n>-ρ e^u_n|ψ_n|^2 ) _g =o(ψ_n).

	Combining these  estimates with (<ref>) we see that
	
    c+o(1)
    	=   ∫_M|∇u_n|^2_g-8π(-1)u̅_n+∫_M e^2u_n_g
    	+2∫_M (<ψ_n,ψ_n>-ρ e^u_n|ψ_n|^2 )_g  
    ≥   ∫_M|∇u_n|^2_g-4π(-1)(2u̅_n-1/1+ε)
    	+1-ε/1+ερ∫_M e^u_n|ψ_n|^2_g
    	 +C(ε,ρ)o(1)+o(ψ_n),

	which is to say,
	
    ∫_M |∇u_n|^2+1-ε/1+ερ e^u_n|ψ_n|^2_g
    	≤ c+4π(-1)(2u̅_n-1/1+ε)+C(ε,ρ)o(1)+o(ψ_n).

	
	Now we estimate the averages u̅_n. 
	Note that by (<ref>) and (<ref>) we also obtain
	
    ∫_M e^2u_n_g 
    	 ≤4π(-1)/1-ε
    			+1+ε/1-ε∫_M ρ e^u_n|ψ_n|^2_g +C(ε,ρ)o(1).

	Then by Jensen's inequality,
	
    e^2u̅_n≤    e^2u̅_n_M e^2u_n_g =1/4π(-1)∫_M e^2u_n_g 
    ≤   1/1-ε+1/4π(-1)1+ε/1-ε∫_M ρ e^u_n|ψ_n|^2_g+ C(ε,ρ,) o(1) 
    ≤   1/1-ε+(1+ε/1-ε)^2 (c/4π(-1)+2u̅_n-1/1+ε)+ C(ε,ρ,) o(1)+ C(ε,) o(ψ_n).

	Thus there exists  C=C(ε,ρ,)>0 such that 
	
    |u̅_n|≤ C(1+c+o(ψ_n)).

	
	The spinors can be controlled by the above growth estimates. 
	Testing (<ref>) against ψ_n^+, we find 
	
    4∫_M ( <ψ_n,ψ_n^+>-ρ e^u_n<ψ_n,ψ_n^+> ) _g
    	-∫_M <φ_n-ρ e^u_nφ_n,ψ_n^+>_g
    	=<β_n,ψ_n^+>_H^-1/2× H^1/2.

	It follows that
	
    Cψ_n^+^2
    	≤   ∫_M<ψ_n,ψ^+>_g
    	    =∫_Mρ e^u_n<ψ_n,ψ_n^+>_g
    	    +1/4∫_M<φ_n-ρ e^u_nφ_n,ψ_n^+>_g+o(ψ_n^+)
    ≤   (∫_M e^u_n|ψ_n|^2_g)^1/2(∫_M e^2u_n_g)^1/4(∫_M|ψ_n^+|^4_g)
       	+φ_nψ_n^+
    			+ρ(∫_M e^2u_n_g)^1/2φ_nψ_n^+ +o(ψ_n^+) 
    ≤    C(1+c+o(ψ_n))^3/4ψ_n^+ + o(ψ_n^+).

 	For what concerns  the other component ψ_n^-, we use (<ref>) to get
 	
    Cψ_n^-^2
     	≤   -∫_M <ψ_n^-,ψ_n>_g
     		=-ρ∫_M e^u_n<ψ_n,ψ_n^->_g 
    ≤   ρ(∫_M e^2u_n_g)^1/4(∫_M e^u_n|ψ_n|^2_g)^1/2ψ_n^-
    ≤    C(1+c+o(ψ_n))^3/4ψ_n^-.

 	Consequently,
 	
    ψ_n^2=ψ_n^+^2+ψ_n^-^2 
     	≤ C(1+c+o(ψ_n))^3/4ψ_n + o(ψ_n).

     Thus there exists some constant C=C(c,,ρ)>0 such that
 	
    ψ_n≤ C(c,,ρ)<+∞.

 	This uniform bound (depending on the level c) in turn gives bounds on  u̅_n and thus
 	
    ∫_M ( |∇u_n|^2+ρ e^u_n|ψ_n|^2 ) _g 
      	≤ C'(c,,ρ)<∞.




Now, passing to a subsequence if necessary, we may assume that there exist u_∞∈ H^1(M) and ψ_∞∈ H^1/2(S) such that

    u_n⇀ u_∞ weakly in  H^1(M), 
       ψ_n⇀ψ_∞ weakly in  H^1/2(S).





 The pair (u_∞,ψ_∞) is a smooth solution of (<ref>). 


 According to the compactness of the Moser–Trudinger embedding (<cit.>), we see that
 
    e^u_n→ e^u_∞ strongly in  L^p(M),     (p<∞).

 Meanwhile, thanks to Rellich-Kondrachov compact embedding Theorem (see e.g. <cit.>)
 
    ψ_n→ψ_∞ strongly in   L^q(S),   (q<4).

 Hence e^u_n|ψ_n|^2 converges weakly in L^p to e^u_∞|ψ_∞|^2, for any p<2.

 It follows that (u_∞, ψ_∞) is a weak solution to (<ref>).
 As remarked before, any weak solution is a classical, hence smooth, solution.  


In particular, this implies that the weak limit (u_∞,ψ_∞) is in the Nehari manifold N.  
Consider the differences

    v_n u_n-u_∞, 
       ϕ_nψ_n-ψ_∞.

Then (v_n,ϕ_n)⇀ (0,0) weakly in H^1(M)× H^1/2(S). 
The functions v_n satisfy

    Δ_g v_n
     =(e^2u_n-e^2u_∞)-ρ(e^u_n|ψ_n|^2-e^u_∞|ψ_∞|^2),

where the right-hand sides converge to 0 in L^q(M) for any q<2.  
This implies that the v_n's are uniformly bounded in H^1(M) and converge strongly to a limit function v_∞∈ H^1(M) satisfying

    Δ_g v_∞=0.

Thus v_∞=const., which has to be zero since v_n⇀ 0. 
This implies the strong convergence of u_n to u_∞ in H^1(M). 
Let us look next at the equations for ϕ_n's:

    _gϕ_n=ρ e^u_nψ_n- ρ e^u_∞ψ_∞,

where the right-hand sides converges to 0 in L^q(S) for any q<4.
Thus there exists  ϕ_∞∈ H^1/2(S) such that ϕ_n converges strongly in H^1/2(S) to ϕ_∞, which satisfies

    _gϕ_∞=0.

By the assumption of trivial kernel on _g, we have that ϕ_∞=0, that is  ψ_n converges strongly to ψ_∞ in H^1/2(S). 
Since N is a submanifold of H^1(M)× H^1/2(S), the sequence (u_n,ψ_n) also converges inside N to (u_∞,ψ_∞); in other words, (u_∞,ψ_∞) lies in the closure of {(u_n,ψ_n)} relative to N. 
Thus we verified the following 



 The functional J_ρ|_N satisfies the Palais-Smale condition. 






§ MOUNTAIN PASS AND LINKING GEOMETRY

In this section we will show that the functional J_ρ|_N, for suitable ρ's, possesses either a mountain pass or linking geometry around the trivial solution (0,0), which will yield existence of a non-trivial min-max critical point.  

For later convenience let us introduce the notation

    F(u)∫_M ( |∇ u|^2-2u+e^2u) _g,        
     Q(u,ψ)2∫_M ( <ψ,ψ>-ρ e^u|ψ|^2 ) _g.

Then we have

 
  (i) J_ρ(u,ψ)=F(u)+Q(u,ψ).
 
  (ii) F(u) ≥ 4π(-1), and this lower bound is achieved by the unique minimizer u_min≡0.  
 
  (iii) Q(u,ψ) is quadratic in ψ and  strongly indefinite. 





 §.§ Local behavior near (0,0)


Let (u,ψ)∈ N be close to (0,0). 
The constraint that defines N, i.e. P^-(1+||)^-1(ψ-ρ e^uψ)=0, implies

    ∫_M<ψ-ρ e^uψ,P^-ψ>_g=0.

Hence we get

    -∫_M<ψ,ψ^->_g
     =   -ρ∫_M e^u<ψ^++ψ^-,ψ^-> 
    
     =   -ρ∫_M e^u|ψ^-|^2_g 
       -ρ∫_M e^u<ψ^+,ψ^->_g.

Since e^u_L^p≤ C(1+u_H^1)≤ C for u uniformly  bounded, 
we have

    ψ^-≤ Cρψ^+.


Now consider the functional

    J_ρ(u,ψ)
         = F(u)+Q(u,ψ)=F(u)+2∫_M ( <ψ,ψ>-ρ e^u|ψ|^2 ) _g 
        = F(u)+2∫_M<(-ρ)ψ, ψ^+>_g
     +2∫_M ρ (1-e^u)<ψ,ψ^+>_g.

The last integral is now of cubic order in (u,ψ), i.e.:

    2∫_M ρ (1-e^u)<ψ,ψ^+>_g
     ≤ Cu_H^1ψψ^+.

For the first term, if we take the equivalent norm |u̅|^2+∇u_L^2^2∼u_H^1^2, then for t^2=|u̅|^2+∇u_L^2^2>0

 
  * if |u̅|^2≥t^2/2≥∇u_L^2^2, then 
       
    F(u)≥∫_M ( e^2u̅-2u̅) _g ≥ 4π(-1)+Ct^2,

 
  * if ∇u_L^2^2≥t^2/2≥|u̅|^2, then 
      
    F(u)≥∫_M ( |∇u|^2+1 ) _g≥4π(-1)+1/2t^2,


thus in either case we have

    F(u)≥ 4π(-1)+C^-1u_H^1^2.

It remains to analyze the middle integral term in the r.h.s. of (<ref>). 
As before, we write ψ=∑_j∈ℤ_* a_jφ_j: 
then

    2∫_M<(-ρ)ψ, ψ^+>_g
     =∑_j>0 2(λ_j-ρ)a_j^2.


From now on we assume that ρ∉(). Thus the above summation can be split into two parts

    2∫_M<(-ρ)ψ, ψ^+>_g
     =-∑_0<λ_j<ρ 2(ρ-λ_j)a_j^2
     +∑_λ_j>ρ 2(λ_j-ρ)a_j^2.




  §.§.§ Mountain pass geometry

First we consider the easier case 0<ρ<λ_1, so the first part of the above summation vanishes. 
Then, locally near (0,0) in  N, we have

    J_ρ(u,ψ) 
     ≥    4π(-1)+C^-1u_H^1^2+ C^-1(1-ρ/λ_1)ψ^+^2-Cu_H^1ψψ^+
    ≥    4π(-1)+C^-1u_H^1^2+ C^-1(1-ρ/λ_1-C^2ψ^2)ψ^+^2,

where we have used Cauchy-Schwarz inequality for the last   term, of cubic order. 
It follows that when
u_H^1^2+ψ_H^1/2^2=r^2>0 is small, there exists a continuous function θ(r)>0 such that

    J_ρ(u,ψ)≥ J(0,0)+θ(r).

On the other hand, we can choose a large constant u̅_1∈ H^1(M) such that ρ e^u̅_1>λ_1+1 and then take s>0 large such that

    J(u̅_1, sφ_1) 
     =   vol(M,g)(e^2u̅_1-2u̅_1) 
      +2(λ_1-ρ e^u̅_1)s^2 
    
     =   4π(-1)(e^2u̅_1-2u̅_1)
      -2(ρ e^u̅_1-λ_1)s^2

is negative. 
Thus we have the mountain pass geometry locally near  (0,0) in the Nehari manifold N. 
Let Γ be the space of paths connecting (0,0) and (u̅_1, sφ_1) inside N (notice that 
Γ≠∅ since N is contractible, and hence connected), parametrized by t∈ [0,1], and define

    c_1inf_α∈Γsup_t∈ [0,1] J_ρ(α(t)).

From the above arguments we have that c_1>4π(-1).
It follows that c_1 is a critical value for J_ρ with a critical point at this level, which is different from the trivial one. 
This concludes the proof of Theorem <ref> in this case. 





  §.§.§ Linking geometry

Next we consider the case ρ∈(λ_k,λ_k+1) for some k≥ 1. 
Now there are more directions in which J_ρ becomes negative, but these are at most finitely-many, and we will apply a linking method to 
exploit the geometry of the functional. 

 Decomposing first the space H^1/2(S) into two parts: 

    H^1/2,k+   {ϕ_1∈ H^1/2(S)|ϕ_1=∑_j>ka_jφ_j}, 
    
     H^1/2,k-   {ϕ_2∈ H^1/2(S)|ϕ_2=∑_j≤ ka_jφ_j},

we have then the orthogonal decomposition 

    H^1/2(S)=H^1/2,k+⊕(H^1/2,k-∩ H^1/2,+(S))
     ⊕ H^1/2,-(S).


Now consider the set

    𝒩_k{0}×(H^1/2,k-∩ H^1/2,+(S))⊂ H^1(M)× H^1/2(S).

It is easy to see that 𝒩_k is a linear subspace inside N, and along this subspace the functional J_ρ is not larger than the minimal critical value:

    J_ρ(0,ϕ_1)
     =4π(-1)-2∑_0<j≤ k (ρ-λ_j) a_j^2 
     ≤ 4π( -1).

For τ > 0 let us consider the following cone around 𝒩_k:

    𝒞_τ(𝒩_k){(u,ψ)∈ N
     | u∈ H^1(M), ψ=ϕ_1+ϕ_2+ψ^-∈ H^1/2,k+⊕ (H^1/2,k-∩ H^1/2,+(S))⊕ H^1/2,-(S), 
    u_H^1+ϕ_1^2+ψ^-^2< τϕ_2^2 }.
  
We claim that for τ suitably chosen this cone contains all the decreasing directions, in the sense that outside the cone we can find a region on which the functional is strictly above the ground state level.

Letting (u,ψ)∈ N∖𝒞_τ(𝒩_k), i.e., with ψ=ϕ_1+ϕ_2 decomposed as above, we have

    u^2_H^1+ϕ_1^2+ψ^-^2≥τϕ_2^2.

By (<ref>), which can be now interpreted as

    ψ^-^2≤ Cρ^2(ϕ_1^2+ϕ_2^2),

we see that 

    ϕ_2^2≤1/τ-Cρ^2(u^2_H^1+(1+Cρ^2)ϕ_1^2).

Moreover, this also implies that 

    u_H^1^2+ ϕ_1^2 ≥ C(u^2 +ψ^2),

for some C=C(ρ,τ)>0. 

 

Then in (<ref>) for the scalar component we have as before the control

    F(u)≥ 4π(-1)+Cu_H^1^2.

For the spinorial part, since ψ=ϕ_1+ϕ_2+ψ^- is an orthogonal decomposition, we have

    Q(u,ψ)
     =   2∫_M<(-ρ)ψ,ψ^+>_g
       +2ρ∫_M (1-e^u)<ψ,ψ^+>_g 
    
     =   2∫_M<(-ρ)ϕ_1,ϕ_1>_g
     +2∫_M<(-ρ)ϕ_2,ϕ_2>_g 
       +2ρ∫_M (1-e^u)<ψ,ϕ_1+ϕ_2>_g 
    ≥   C(1-ρ/λ_k+1)ϕ_1^2
         -C(ρ/λ_k-1)ϕ_2^2 
         -Cρu_H^1ψ(ϕ_1+ϕ_2).

Assuming u^2_H^1+ψ^2=r^2 is small and noting (<ref>), we get

    J_ρ(u,ϕ_1+ϕ_2)
     ≥    4π(-1)+Cu_H^1^2
     + C(1-ρ/λ_k+1-Cr^2)ϕ_1^2
        -C(ρ/λ_k-1-Cr^2)ϕ_2^2 
    ≥   4π(-1)+Cu_H^1^2
     + C(1-ρ/λ_k+1-Cr^2)ϕ_1^2
        -C(ρ/λ_k-1)u_H^1^2+(1+Cρ^2)ϕ_1^2/τ-Cρ^2.

We can first choose r small enough and then choose τ large enough such that

    J_ρ(u,ϕ_1+ϕ_2)
     ≥    4π(-1)+ C(u^2+ϕ_1^2) 
    ≥    4π(-1)+C r^2

outside the cone 𝒞(𝒩_k). 
Thus the claim is confirmed. 



For r as above, consider the set

    L_1(  B_r(0,0)\𝒞(𝒩_k)) ∩ N,

which is non-empty since (0,rφ_k+1)∈ L_1.
Recall that N is locally modeled by a Hilbert space, e.g. T_(0,0) N. 
We can assume that in a local chart, 𝒩_k is some coordinate subspace, while L_1 is homeomorphic to a collar neighborhood of the sphere (of infinite dimension) which lies in a subspace complementary to 𝒩_k and intersects 𝒞_τ(𝒩_k) only at {(0,0)}. 

Next we introduce a set L_2 on which the functional attains low values and such that it links with L_1, see Figure <ref>. 
The construction of such a set is performed in several steps. First we take the ball

    B_R^0,k(0){(0,ϕ_2)∈𝒩_k |ϕ_2≤ R}⊂𝒩_k

with R>0 a large constant to be fixed later. 
Note that for any (0,ϕ_2), J_ρ(0,ϕ_2)≤ 4π(-1) and for (0,ϕ_2)∈ B_R^0,k(0), 

    J_ρ(0,ϕ_2)
     ≤ 4π(-1)-C(ρ/λ_k-1)ϕ_2^2
     ≤ 4π(-1)-C(ρ/λ_k-1)R^2.

For any (0,ϕ_2)∈ B_R^0,k(0), we consider the following curves. 
First let

    σ_1 [0,T]→ N,   σ_1(t)(t, ϕ_2+Atφ_k+1),

where A>0 is again a constant to be fixed later.
One easily sees that this is a curve in N and 

    J_ρ(t,ϕ_2+   Atφ_k+1)
     =vol(M,g)(e^2t-2t)+2∫_M<(-ρ e^t)ϕ_2,ϕ_2>_g
               +2A^2t^2∫_M<(-ρ e^t)φ_k+1,φ_k+1>_g
    ≤   4π(-1)(e^2t-2t)
     +2∫_M<(-ρ)ϕ_2,ϕ_2>_g 
     +2A^2t^2 (λ_k+1-ρ e^t) 
    ≤   4π(-1)(e^2t-2t)
     -C(ρ/λ_k-1)R^2
     +2A^2t^2 (λ_k+1-ρ e^t).

Now we fix some constants:

 
  * we choose T>0 such that ρ e^T-λ_k+1≥ 1;
 
  * then we choose A>0 such that
       
    4π(-1)(e^2T-2T)-2A^2T^2(ρ e^T-λ_k+1)<4π(-1);

  
  * finally, choose R>0 such that for any t∈ [0,T]
        
    4π(-1)(e^2t-2t)
     -C(ρ/λ_k-1)R^2
     +2A^2t^2 (λ_k+1-ρ e^t)<4π(-1).


Then we consider the curve

    σ_2 [-1,1]→ N,   σ_2(r)(T,(-r)ϕ_2+ATφ_k+1),
 
which joins (T,ϕ_2+ATφ_k+1) to (T,-ϕ_2+ATφ_k+1) inside N. 
Thereafter we can come back to 𝒩_k via the curve

    σ_3 [0,T]→ N,   σ_3(t)((T-t), ϕ_2+A(T-t)φ_k+1).

Finally, consider the subset

    𝒟{(t,Atφ_k+1+ϕ_2) | t∈ [0,T], (0, ϕ_2)∈ B_R^0,k(0) },
 
which is compact and  homeomorphic to a finite-dimensional cylindrical segment 

    [0,T]× B_R^0,k(0).

Note that 𝒟⊂ N and
let L_2=𝒟, see Figure <ref>.    
The curves σ_1, σ_2, σ_3 constructed above pass through every point of L_2\ B_R^0,k(0). It follows that on L_2 the functional attains low values.  
One can shrink L_2 (in an homotopically equivalent way) into the coordinate chart to see that L_1 and L_2 actually link,  see e.g. <cit.> for a rigorous definition of this concept. 






Now we define the linking level.
Let Γ be the space of continuous maps

    α𝒟→ N,

such that α(v,h)=(v,h) for any (v,h)∈ L_2=𝒟. 
This set Γ is clearly non-empty since _𝒟∈Γ. 
Then we define the linking level

    c_1inf_α∈Γmax_(v,h)∈𝒟J_ρ(α(v,h)) .

As L_1 and L_2 link, we have from the above arguments that

    c_1 ≥ 4π(-1)+θ(r).

It follows that c_1 is a critical value for J_ρ, and again we obtain a critical point for J_ρ which is different from the trivial one.        
This concludes the proof of Theorem <ref> in this case as well.
  

HKW


super1
Changrim Ahn, Chaiho Rim, Marian Stanishkov. Exact one-point function of N=1 super-Liouville theory with boundary. Nucl. Phys. B 636 (2002), 497–513.

ambrosetti2007nonlinear
Antonio Ambrosetti, Andrea Malchiodi.
Nonlinear Analysis and Semilinear Elliptic Problems. 
Cambridge Studies in Advanced Mathematics, 104. Cambridge University Press, Cambridge, 2007.

ammann2003habilitation
Bernd Ammann.
A variational problem in conformal spin geometry.
Habilitation, Universität Hamburg, 
2003.


aubin1998somenonlinear
Thierry Aubin.
Some nonlinear problems in Riemannian geometry. 
Springer, Berlin,
1998. 

bar1992harmonic
Christian Bär, Paul Schmutz.
Harmonic spinors on Riemann surfaces.
Annals of Global Analysis and Geometry 10 (1992), 263–273.

benci1982oncritical
Vieri Benci.
On critical point theory for indefinite functional in the presence of symmetries.
Trans. AMS 274 (1982), no. 2, 533–572.

benci1979critical
Vieri Benci, Paul H. Rabinowitz.
Critical points for indefinite functionals.
Invent. Math. 52 (1979), 241–273.

bores1994harmonic
Jarolím Bores.
Harmonic spinors on Riemann surfaces. 
Proceedings of the Winter School "Geometry and
Physics". 
Circolo Matematico di Palermo, Palermo,
Rendiconti del Circolo Matematico di Palermo, Serie II, Supplemento No. 37: 15–32,
1994.

chai-lin-wang
Ching-Li Chai, Chang-Shou Lin, Chin-Lung Wang. Mean field equations, hyperelliptic curves, and modular forms: I. Camb. J. Math. 3 (2015), no. 1-2, 127–274.

changyang1987prescribing
Sun-Yung Alice Chang, Paul C. Yang. 
Prescribing Gaussian curvature on S^2.
Acta Math. 159, (1987), 215–259.

chen-kuo-lin
Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin. Hamiltonian system for the elliptic form of Painlevé VI equation. J. Math. Pure App. 106 (2016), no. 3, 546–581.

super2
Takeshi Fukuda, Kazuo Hosomichi. Super-Liouville theory with boundary. Nucl. Phys. B 635 (2002), 215–254.

ginoux2009dirac
Nicolas Ginoux.
The Dirac Spectrum.
Springer, Berlin,
2009.

gilbarg2001elliptic
David Gilbarg, Neil S. Trudinger. 
Elliptic partial differential equations of second order.
Springer-Verlag, Heidelberg,
2001. 

glockner2016fundamentals
Helge Glöckner.
Fundamentals of submersions and immersions between infinite-dimensional manifolds.
arXiv: 1502.05795v4[math.DG].

hitchin1974harmonicspinors
Nigel Hitchin.
Harmonic spinors.
Adv. Math. 14 (1974), 1–55.

hulshof1993differential
Josephus Hulshof, Robertus Van der Vorst. 
Differential system with strongly indefinite variational structure.
J. Funct. Anal. 114 (1993), 32–58.

isobe2010existence
Takeshi Isobe.
Existence results for solutions to nonlinear Dirac equations on compact spin manifolds.
Manuscript Math. 35 (2011), 329–360.

isobe2011nonlinear
Takeshi Isobe.
Nonlinear dirac equations with critical nonlinearities on compact spin manifolds.
J. Funct. Anal. 260 (2011), 253–307.

isobe2019onthemultiple
Takeshi Isobe.
On the multiple existence of superquadratic dirac-harmonic maps into flat tori.
Calc. Var. PDEs, (2019) 58:126.  

jost2007superLiouville
Jürgen Jost, Guofang Wang, Chunqin Zhou.
Super-Liouville equations on closed Riemann surfaces. 
Comm. PDEs 32 (2007), no. 7, 1103–1128.

jost2009energy
Jürgen Jost, Guofang Wang, Chunqin Zhou, Miaomiao Zhu. 
Energy identities and blow-up analysis for solutions of the super-Liouville equation. 
 J. Math. Pures Appl. 92 (2009), no. 3,  295–312.

jost2014qualitative
Jürgen Jost, Chunqin Zhou, Miaomiao Zhu. 
The qualitative boundary behavior of blow-up solutions of the super-Liouville equations. 
J. Math. Pures Appl. 101 (2014), no. 5, 689–715. 


jost2015LocalEstimate
Jürgen Jost, Chunqin Zhou, Miaomiao Zhu.
A local estimate for super-Liouville equations on closed Riemann surfaces. 
 Calc. Var. PDEs  53 (2015), no. 1-2, 247–264.
 
jost2016regularity
Jürgen Jost, Enno Keßler, Jürgen Tolksdorf, Ruijun Wu, Miaomiao Zhu.
Regularity of solutions of the nonlinear sigma model with gravitino.
Comm. Math. Phys. 358 (2018) no. 1, 171–197. 
 
jost2018symmetries
Jürgen Jost, Enno Keßler, Jürgen Tolksdorf, Ruijun Wu, Miaomiao Zhu.
Symmetries and conservation laws of a nonlinear sigma model with gravitino.
J. Geom. Phys. 128 (2018), 185–198.

kazdan1975scalar 
Jerry L. Kazdan, Frank W. Warner.
Curvature functions for compact 2-manifolds. Ann. Math. 99 (1974), 14–47. 


lawson1989spin
H. Blaine Lawson, Jr. ,Marie-Louise Michelsohn.
Spin geometry.
Princeton Uni. Press,
1989.

maalaoui2013rabinowitz
Ali Maalaoui.
Rabinowitz-Floer Homology for super-quadratic Dirac equations on spin manifolds.
J. Fixed Point Theory Appl. 13 (2013), 175–199.

maalaoui2015therabinowitz
Ali Maalaoui, Vittorio Martino.
The Rabinowitz–Floer homology for a class of semilinear problems and applications.
J. Funct. Anal. 269 (2015), 4006–4037.

maalaoui2017characterization
Ali Maalaoui, Vittorio Martino.
Characterization of the Palais-Smale sequences for the conformal Dirac-Einstein problem and applications.
J. Diff. Eq. 266 (2019), 2493–2541.

ops1988extremals  
Brad Osgood, Ralph Phillips, Peter Sarnak. 
Extremals of determinants of Laplacians. 
J. Funct. Anal. 80 (1988), 1, 148–211. 

pankov2005periodic
Alexander Pankov.
Periodic nonlinear Schrödinger equation with application to photonic crystal. 
Milan J. Math. 73 (2005), 259–287.

polya1
Aleksandr M. Polyakov. Quantum geometry of fermionic strings. Phys. Lett. B 103 (1981), 207–210. 

polya2
Aleksandr M. Polyakov. Gauge Fields and Strings. Contemporary concepts in physics, Chur, Switzerland: Harwood Academic Publishers, 1987.

super3
João N.G.N. Prata. The super-Liouville-equation on the half-line. Nucl. Phys. B 405 (1997), 271–279.


schoenyau1994lectureOnDG-I
Richard Schoen, Shing-Tung Yau.
Lectures on differential Geometry. 
International Press of Boston, Inc,
1994.

hawking
Yuguang Shi, Jiacheng Sun, Gang Tian, Dongyi Wei. Uniqueness of the mean field equation and rigidity of Hawking Mass. Calc. Var. and PDEs (2019), 58:41.

spruck-yang
Joel Spruck, Yisong Yang. On Multivortices in the Electroweak Theory I: Existence of Periodic Solutions. 
Comm. Math. Phys. B (1992), 1–16.

struwe2008variational
Micheal Struwe.
Variational methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. 
Springer-Verlag, Berlin Heidelberg,
2008.

szulkin2009ground
Andrzej Szulkin, Tobias Weth.
Ground state solutions for some indefinite variational problems.
J. Funct. Anal. 257 (2009), 3802–3822.

szulkin2010themethod
Andrzej Szulkin, Tobias Weth.
The method of Nehari manifold.
Handbook of Nonconvex Analysis and Applications.
Int. Press, Somerville, MA, (2010), 597–632.


tarantello
Gabriella Tarantello. Multiple condensate solutions for the Chern-Simons-Higgs theory. 
J. Math. Phys. 37 (1996), 3769–3796.

yang
Yison Yang. Solitons in Field Theory and Nonlinear Analysis. Springer Monographs in Mathematics, Springer, New York, 2001.

  
