YANG Xu(楊旭) ,LI Xin(李鑫)

( 1.School of Mathematics , Yunnan Normal University, Kunming 650500, China;2.Yunnan Key Laboratory of Modern Analytical Mathematics and Applications,Kunming 650500, China)

Abstract: In this article we study the existence of solutions for the Dirac systems with the chirality boundary condition.Using an analytic framework of proper products of fractional Sobolev spaces,the solutions existence results of the coupled Dirac systems are obtained for nonlinearity with superquadratic growth rates.The results obtained by GONG and LU (2017) are extended to the case of chiral boundary condition.

Key words: Dirac system;Boundary condition;Variational method

1.Introduction and Main Results

Dirac operators on compact spin manifolds play prominent role in the geometry and mathematical physics,such as the generalized Weierstrass representation of the surface in three manifolds[9]and the supersymmetric nonlinear sigma model in quantum field theory[4].The existence of solutions of the Dirac equation has been studied on compact spin manifolds without boundaries by Ammann[1],Isobe[12].In addition,being different with these existing works,DING and LI[5]studied a class of boundary value problem on a compact spin manifoldMwith smooth boundary.The problem is a general relativistic model of confined particles by means of nonlinear Dirac fields onM.In this paper,we are concerned with a nonlinear Dirac systems on compact spin manifolds with smooth boundary,and deal with some existence results.

Spin manifold (M,g) equipped with a spin structureσ:Pspin(M)→Pso(M),and let ΣM=Pspin(M)×σΣmdenote the complex spinor bundle onM,which is a complex vector bundle of rank 2[m/2]endowed with the spinorial Levi-Civita connection?and a pointwise Hermitian scalar product〈·,·〉.We always assumem ≥2 in this paper.Consider Whitney direct sum ΣM ⊕ΣMand write a point of it as (x,u,v),wherex ∈Mandu,v ∈ΣxM.Pis the Dirac operator under the boundary conditionBCHIu=BCHIv=0 on?M.We consider the following system of the coupled equations

where fiber preserving mapH: ΣM ⊕ΣM →R is a real valued superquadratic function of classC1with subcritical growth rates.(1.1) is the Euler-lagrange equation of the functional

where dxis the Riemann volume measure onMwith respect to the metricg,〈·,·〉 is the compatible metric on ΣM.

The problem (1.1) describes two coupled fermionic fields in quantum.It can be viewed as a spinorial analogue of other strongly indefinite variational problems such as elliptic systems[2,8,11].A typical way to deal with such problems is the min-max method,including the mountain pass theorem,linking arguments.Another is a homological method,the Morse theory and Rabinowitz-Floer homology.For the Dirac operator associated with appropriate boundary condition,Farinell and Schwarz[7]prove that Dirac operatorPis elliptic and extends to a self-adjoint operator with a discrete spectrum.In this paper we use the techniques introduced by Hulshof and Van Der Vorst[11]to prove the existence of solutions of(1.1),and apply a generalized fountain theorem established by Batkam and Colin[2]to obtain infinitely many solutions of the coupled Dirac system provided the nonlinearityHis even.In the following we assume that two real numbersp,qsatisfy

For the nonlinearityH,we make the following hypotheses:

H ∈C0(ΣM ⊕ΣM,R) isC1in the fiber direction.Real constants 2

(H1) There exist a constantC1>0 such that

(H2) There existR1>0,such that

for all (x,u,v)∈ΣM ⊕ΣMwith|(u,v)|≥R1.

(H3)H(x,u,v)≥0 for all (x,u,v)∈ΣM ⊕ΣM.

(H4)H(x,u,v)=o(|(u,v)|2) as|(u,v)|→0 uniformly forx ∈M.

(H5)H(x,-u,-v)=H(x,u,v) forany (x,u,v)∈ΣM⊕ΣM.

Note1.1ThatH(x,u,v)=|u|p+|v|qsatisfies these conditions.In[15]existence results for the Dirac system without boundary condition are given under the same assumptions onH(x,u,v).The above condition (H2) looses associated condition in [10].

Our main result is as follow.

Theorem 1.1If the aboveHsatisfies (H1)-(H4),then the Dirac system(1.1) possesses at least one solution.

Furthermore,for odd nonlinearities we have the following multiplicity result:

Theorem 1.2If the aboveHsatisfies (H1)-(H5),then there exists a sequence of solutionsto (1.1) with L(uk,vk)→∞ask →∞.

2.About Boundary Condition

We collect here some basic definitions and facts about spin structures on manifolds and Dirac operators.For more detailed exposition,please consult [5,13].

Define(ΣM,〈,〉,γ,?)is a Dirac bundle if?:C∞(M,ΣM)→C∞(M,T?M ?ΣM)andγ:C∞(M,T?M ?ΣM)→C∞(M,ΣM) satisfies:

for anyω ∈TMand?,ψ ∈C∞(M,ΣM).We have used the identificationT?M ～=TMby the metric onM,thenC∞(M,T?M ?ΣM)～=C∞(M,TM ?ΣM),therefore,Dirac operatorPact on spinors onMis defined by

In particular,if we choose an local orthogonal tangent frame{e1,e2,···,em},the Dirac operatorPbecomes

Then we consider a Chirality operator associated with the Dirac bundle〈ΣM,〈·〉,γ,?〉.If a linear mapF:EndC(ΣM)→EndC(ΣM)satisfies

for each vector fieldX ∈TMand spinor fieldsψ,φ ∈C∞(M,ΣM).

The boundary hypersurface?Mis also a spin manifold and so we have the corresponding spinor bundle Σ?M,the clifford multiplicationγ?M,the spin connection??Mand the intrinsic Dirac operatorP?M.The restricted Hermitian bundle ΣM|?Mcan be identified with the intrinsic Hermitian spinor bundle Σ?M,provided thatmis odd.Instead,ifmis even,the restricted Hermitian bundle ΣM|?Mcould be identified with the sum Σ?M ⊕Σ?M.

Define an operatorΓ:=F|?M γ(N),whereNdenotes the unite inner normal vector field on?M.By the definition,we knowFis a local operator on the spinor bundle over?M.Fis a self-adjoint operator and has two eigenvalues +1 and-1.The corresponding eigenspaces are

For the spaceC∞(M,ΣM),defineaninnerproduct

ThenH1(M,ΣM) is the completion of the spaceC∞(M,ΣM) with respect to the norm∥·∥H1.SincePis a first operator,it extends to a linear operatorP:H1(M,ΣM)→L2(M,ΣM) andP|?M:H1(M,ΣM)→L2(?M,Σ?M).Let

Then the Dirac operatorPwith Chirality boundary conditionBCHIψ|?M=0 is well defined in the domain D(P).For simplicity,in the following,we will denote the D(P) by D.

Forψ,φ ∈D,by the integrated version of Lichnerowitz Formula,we have (Pψ,φ)=(ψ,Pφ).Actually,Pis a self-adjoint operator inL2(M,ΣM) with domain D.

3.The Analytic Framework

If (M,g) has positive scalar curvature,it is obviously 0Spec(P),by Fridrich’s inequality.

is a Hilbert space isomorphism by the arguments above.Hence

It is a self-adjoint isometry operator andB?B=Id:E →Eis identity operator.Introducing the “diagonals”

Note thatBz±=±z±,soE+andE-are the mutually orthogonal eigenspaces of the eigenvalues 1 and-1 ofB.Orthonormal bases consisting of eigenvectors ofE±are given by

Then for eachz=z++z-,we have

Now we can define a functionalL:E →R as

whereH(z)=H(x,z)dx.

SinceMis compact,by the assumption (H1) and integrating we obtain

And letu=0.Similarly,we prove that

From (3.2),(3.3) and Young’ inequality to derive

for some constantC>0.

By an analysis of interpolation of the Sobolev spaces,

Since (D,∥·∥1,2) embedsH1(M,ΣM) continuously,there holds the continuous embeddingThen using (3.4) we can define the functionalH:E →R as

is of classC1and its derivative at (u,v)∈Eis given by

MoreoverDH:E →E?is a compact operator.

In fact,using the H¨older inequality and embeddings we have

In a similar way we obtain an inequality for the derivative with respect tov.ThusDH(u,v)is well defined and bounded inE.Next,by the Sobolev embeddings,usual arguments give thatDH(u,v) is compact.

4.The Palais-Smale Condition for L

LetFbe aC1functional on a Banach spaceE,c ∈R.Recall that a sequence{xn}?Eis called a (PS)c-sequence ifF(xn)→casn →∞and∥DF(xn)∥E?→0 asn →∞.If all(PS)c-sequences converge inE,we say thatFsatisfies the (PS)ccondition.In this section we prove the (PS)ccondition for L.

Lemma 4.1SupposeHsatisfies (H1),(H2).Then for anyc ∈R,L satisfies the (PS)ccondition with respect toE.

ProofLet{zn}={(un,vn)} ?Ebe a (PS)c-sequence with respect toE,i.e.,zn ∈Eand satisfy

Claim 4.1{zn}?Eis bounded.

The condition (H2) implies that there are constantsC2,C3>0 such that

See [6] for a proof.By (4.1)-(4.3) and (H2),for largenwe have

Using the conditions (iii) an d (iv) above (H1),an analogous reasoning yields

Moreover,it also holds that

By (4.7)-(4.12),we deduce

Hence (4.13) and (4.5) lead to

For anyz-∈E-,then the similar arguments will lead to

Adding (4.14) and (4.15) yields

By the assumptions onp,q,μabove (H1),it is easily checked that the total exponent each term in the right-hand side of (4.16) is less than 2.It follows that the sequence{zn}is bounded inE.Claim 4.1 is proved.

Passing to a subsequence we may assume that for somez ∈E,zn ?zweakly inE.From here on a usual argument based on the compactness ofDHand invertibility ofBgive the existence of a subsequence ofzn=B-1(DL(zn)+DH(zn)) that convergeszinE.So the(PS)c-condition is verified.

5.Proof of the Theorems

The proof of Theorem 1.1 is based on an application of the following theorem of Benci and Rabinowtitz[3].

Theorem 5.1(Indefinite Functional Theorem) LetHbe a real Hilbert space withH=H1H2.satisfies the Palais-Smale conditon,and

(I1)L(z)=(Lz,z)-H(z),whereL:H →His bounded and self-adjoint,andLleavesH1andH2invariant;

(I2)DHis compact;

(I3) there exists a subspace?Hand setsS ?H,Q ?and constantsα>ωsuch that

(i)S ?H1andL|S≥α,

(ii)Qis bounded and L≤ωon the boundary?QofQ ∈

(iii)Sand?Qlink,then L possesses a critical valuec ≥α.

Before giving the geometric conditions for the first linking property,we sets1,s2,ρ>0 with 0<ρ

whereBρdenotes an open ball with radiusρcentered at the origin,e+=(ξ+,η+)∈E+withη+some eigenspinor ofPcorresponding to the first positive eigenvalue

Lemma 5.1There existsρ>0 andα>0 such that

ProofConditons (H1),(H3) and (H4) imply that for anyε>0 there exists a constantC(ε)>0 such that

for all (u,v)∈E.Combining (5.1) and the Sobolev embedding,it is straightforward to show that

for some constantsC4>0 andC5>0.Thus we can fixε0,α>0 such that L(z+)≥αonS.

Lemma 5.2There existss1,s2,ρ>0 with 0<ρ

ProofNote that the boundary?Qof the cylinderQis taken in the spaceand consists of three parts,namely the bottomQ∩{s=0},the lidQ∩{s=s1},and[0,s1e+]⊕(∩E-).

Clearly L(z)≤0 on the bottom by (H3).For the remaining two parts of the boundary we first observe that,forz=z-+re+∈

By definition ofE+we haveξ+=|P|-1Pη+=η+,therefore,e+=(η+,η+).

We setz-=(u-,v-),forz-+re+=(u-+rη+,v-+rη+).Using (4.3),we have

Thus,writingv-=tη++,whereη+is orthogonal toinL2(M,ΣM).By definition ofE±we have

Similarly,η+is orthogonal to|P|-1inL2(M,ΣM).By H¨older’s inequality,

for some constantC6depending onη+.Similarly,we have

Therefore,we deduce from (5.3),(5.4) and (5.5)that

Byμ>2,takingr=s1large enough we see in (5.6) that L(z-+re+)<0 on the lidQ ∩{s=s1}.

Forz-+re+∈[0,s1e+]⊕(?Bs2∩E-),we deduce from the condition (H3) that

Taking∥z-∥E=s2large enough,it holds that

The desired result is proved.

Proof of Theorem 1.1LetH=E,H1=E+,H2=E-,we apply Lemma 4.1 to the functional L.The Palais-Smale conditon is satisfied.We can use the standard methods to show that Conditions I1,I2and L is continuously differentiable.The geometric conditions I3(i),(ii) is proved in Lemma 5.1 and Lemma 5.2.For the proof of (I3)(iii) we refer to [3].Therefore L possesses a critical value pointz ∈Eand satisfies L(z)≥α>0.

To obtain Theorem 1.2,we recall the Generalized fountain theorem for semi-definite functionals (see [6] for the detailed exposition).

(A3) L satisfies the Palais-Smale condition;where

Then L has an unbounded sequence of critical values.

We will define the following subsets for giving the geometric conditions of the linking property:

Bk:={z ∈Yk|∥z ∥E≤ρk},Nk:={z ∈Zk|∥z ∥E=rk},?Bk:={z ∈Yk|∥z ∥E=ρk},where 0

Lemma 5.3There existsρk>rk>0 such that

(A1)ak:=L(z)→∞,k →∞;

(A2)bk:=L(z)≤0 anddk:=L(z)<∞.

Proof(i) Letz=(u,v)∈Zk,T=max{p,q},t=min{p,q},Then by (5.1),which implies that

We know by Lemma 3.8 in [14] thatαk →0 ask →∞,so L(z)→∞ask →∞,and the condition (A1) is satisfied.

SinceE+is orthogonal toE-inL2(M,ΣM ⊕ΣM),we deduce

Combining (3.1) with (5.9) yields

Takingδ>shows that L(z)→-∞as∥z ∥E→∞,so (A2) is satisfied forρklarge enough.The desired result is proved.

LetΠ-:E →E-andΠ+:E →E+be the orthogonal projections,be an orthonormal basis ofE-.OnEwe consider a new norm

We use theτ-topology is generated by the norm||||·||||[14].It is clear that∥Π+z ∥E≤||||z||||≤∥z ∥E.Moreover,if{zn}is a bounded sequence inEthen

Lemma 5.4Under the assumptions (H1) and (H3),L isτ-upper semicontinuous andDL is weakly sequentially continuous.

This shows thatDL is weakly sequentially continuous.

Now we use Theorem 5.2 to obtain infinitely many critical points of the even functional L in Theorem 1.2.

Proof of Theorem 1.2We know by Lemma 4.1 that L satisfies the Palais-Smale condition.From Lemma 5.3,it can be seen that the geometric conditions (A1) and (A2)hold.Lemma 5.4 implies that L isτ-upper semicontinuous andDL is weakly sequentially continuous.Then L has an unbounded sequence of critical values.