ON COUPLED DIRAC SYSTEMS

. In this paper, we show the existence of solutions for the coupled Dirac system on M, where M is an n -dimensional compact Riemannian spin manifold, D is the Dirac operator on M , and H : Σ M ⊕ Σ M → R is a real valued superquadratic function of class C 1 in the ﬁber direction with subcritical growth rates. Our proof relies on a generalized linking theorem applied to a strongly indeﬁnite functional on a product space of suitable fractional Sobolev spaces. Further-more, we consider the Z 2 -invariant H that includes a nonlinearity of 1 , where f ( x ) and g ( x ) are strictly positive continuous functions on M and p,q > n . In this case we obtain inﬁnitely many solutions of the coupled Dirac system by using a generalized


(Communicated by Kuo-Chang Chen)
Abstract. In this paper, we show the existence of solutions for the coupled Dirac system where M is an n-dimensional compact Riemannian spin manifold, D is the Dirac operator on M , and H : ΣM ⊕ ΣM → R is a real valued superquadratic function of class C 1 in the fiber direction with subcritical growth rates. Our proof relies on a generalized linking theorem applied to a strongly indefinite functional on a product space of suitable fractional Sobolev spaces. Furthermore, we consider the Z 2 -invariant H that includes a nonlinearity of the form where f (x) and g(x) are strictly positive continuous functions on M and p, q > 1 satisfy 1 p + 1 + 1 q + 1 > n − 1 n .
In this case we obtain infinitely many solutions of the coupled Dirac system by using a generalized fountain theorem.
inner product (u, v) L 2 = M u(x), v(x) dx on the space C ∞ (M, ΣM ) of all C ∞sections of the bundle ΣM , where dx is the Riemannian measure of g. Denote by L 2 (M, ΣM ) the space of L 2 -sections of ΣM . The Dirac operator is an elliptic differential operator of order one, D = D g : C ∞ (M, ΣM ) → C ∞ (M, ΣM ), locally given by Dψ = n i=1 e i ·∇ ei ψ for ψ ∈ C ∞ (M, ΣM ) and a local g-orthonormal frame {e i } n i=1 of the tangent bundle T M . Consider Whitney direct sum ΣM ⊕ ΣM of ΣM and itself, and write a point of it as (x, ξ, ζ), where x ∈ M and ξ, ζ ∈ Σ x M . In this paper we study the following system of the coupled semilinear Dirac equations: where u(x), v(x) ∈ Σ x M ∀x ∈ M and H : ΣM ⊕ ΣM → R is C 1 in the fiber direction. (1) is the Euler-Lagrange equation of the functional The functional L H is strongly indefinite since the spectrum of the operator D is unbounded from below and above. Nonlinear Dirac equations arise in many interesting problems in geometry and physics, such as the generalized Weierstrass representation of surfaces in threemanifolds [15] and the supersymmetric nonlinear sigma model in quantum field theory [10,11]. In [2,3] Ammann studied a class of Dirac equations of the form Dψ = λ|ψ| p−1 ψ for λ > 0 and 2 < p ≤ 2n/(n − 1). Raulot [24] extended Ammann's results to the equations of the form Dψ = H(x)|ψ| p−1 ψ, where H is a smooth positive function whose all partial derivatives at some maximum point of order less than or equal to n − 1 vanish. Furthermore, Isobe [19] studied super-linear and sub-linear nonlinear Dirac equations on compact spin manifolds. Besides, Isobe [20] used the dual variational method to obtain the existence of nonlinear Dirac equations with critical nonlinearities. Recently, in [17] the authors extended Isobe's results to a large class of nonlinear Dirac equations with positive C 1 -potentials.
However, except for these works, the coupled Dirac systems like (1) are less studied in the literatures. In quantum physics, the problem (1) describes two coupled fermionic fields. Mathematically, it can be viewed as a spinorial analogue of other strongly indefinite variational problems such as infinite dynamical systems [5,6] and elliptic systems [4,12], and this is our main motivation for its study. A typical way to deal with such problems is the min-max method of Benci and Rabinowitz [9,23], including the mountain pass theorem, linking theorem and so on. In this paper we use the techniques introduced by Felmer [13] to prove the existence of solutions of (1), and apply a generalized fountain theorem established by Batkam and Colin [8] to obtain infinitely many solutions of the coupled Dirac system provided the nonlinearity H is even.
In the following we assume that two real numbers p, q > 1 satisfy It is not hard to verify that we can choose a real number s ∈ (0, 1) such that p < n + 2s n − 2s and q < n + 2 − 2s n + 2s − 2 .
On nonlinearity H, we make the following hypotheses: (H3) There exists a constant c 1 > 0 such that (H4) H(x, u, v) = o(|u| 2 + |v| 2 ) uniformly with respect to x as |u| + |v| → 0. Consider the following typical examples satisfying the above (H1)-(H4), where f (x) and g(x) are strictly positive continuous functions on M . Then (1) reduces to the following form Note that M H(x, u, v)dx is not well-defined on the Hilbert space H 1 2 (M, ΣM ) × H 1 2 (M, ΣM ) unless we make a stronger hypothesis on the exponents p, q as in [12]. To overcome this difficulty, inspired by the ideas of Hulshof and Van der Vorst [18], we consider the following well-defined functional For even nonlinearities, we have the following multiplicity result: Theorem 1.2. Assume that n ≥ 2 and 0 / ∈ Spec(D). If H : ΣM ⊕ ΣM → R satisfies (H1)-(H4). Furthermore, assume H is even in the fiber direction, i.e., A direct application of Theorem 1.2 is the following corollary: Corollary 1. Let H be as in (8). Assume that n ≥ 2 and 0 / ∈ Spec(D). Then (9) has infinitely many solutions Organization of the paper. In Section 2, we define a functional on a suitable product space of fractional Sobolev spaces and give some regularity results on the functional A H and solutions of (1) obtained by its critical points. The aim of Section 3 is to prove the Palais-Smale condition. In Section 4, we given some geometric conditions of link argument. The proofs of Theorem 1.1 and Theorem 1.2 are given in Sections 5, 6 respectively.
2. The analytic framework. Let (M, g) be as in Section 1. The Dirac operator and its spectrum consists of an unbounded sequence of real numbers (cf. [14,22]). The well known Schrödinger-Lichnerowicz formula implies that all eigenvalues of D are nonzero if M has positive scalar curvature. Hereafter, we assume: , the volume of (M, g) equals to 1.
(The second assumption is only for simplicity, it is actually unnecessary for our result!). Let (ψ k ) ∞ k=1 be a complete L 2 -orthonormal basis of eigenspinors corresponding to the eigenvalues (λ k ) ∞ k=1 counted with multiplicity such that |λ k | → ∞ as k → ∞. For each s ≥ 0, let H s (M, ΣM ) be the Sobolev space of fractional order s, its dual space is denoted by H −s (M, ΣM ). We have a linear operator |D| s : ) is compact and self-adjoint. |D| s can be used to define a new inner product on H s (M, ΣM ), The induced norm · s = (·, ·) s,2 is equivalent to the usual one on H s (M, ΣM ) (cf. [1,2]). For r ∈ R consider the Hilbert spacē Then H s (M, ΣM ) can be identified with the Hilbert spaceω 2s . Hence can be identified withω −2s , where the pairing betweenω −2s andω 2s is given by It follows that |D| −2s gives a Hilbert space isomorphism from H −s (M, ΣM ) to H s (M, ΣM ) with respect to the equivalent new inner products as in (12). Moreover we have a continuous inclusion L 2 (M, ΣM ) → H −s (M, ΣM ) and Consider the Hilbert space with inner product is a Hilbert space isomorphism by the arguments above (14) and Since M is compact, by the assumption (H2) we have constants and by the assumption (H3) we can use Young's inequality to derive for some constant C > 0. (Later on, we also use C to denote various positive constants independent of u and v without special statements). From (18) and (19) we see that the nonlinearity H is superquadric.
Proposition 1. Assume that H ∈ C 0 (ΣM ⊕ ΣM ) satisfies (H1) and (H3). Then the functional H : E s → R defined by is of class C 1 , its derivation at (u, v) ∈ E s is given by and DH : E s → E s is a compact map, where E s consists of all bounded linear functionals on E s .

Remark 1.
If the real numbers p, q satisfy for some s ∈ (0, 1), which implies (3), then the above space E s can be replaced by For the sake of completeness we shall give the proof of Proposition 1 in Appendix A.
It follows from Proposition 1 that the functional A H in (10) is of class C 1 on Hilbert space E s with inner product Since the operator D is self-adjoint, the functional A H can be written as where Note that M Lz(x), z(x) dx = (Lz, z) 2 = (D s Lz, z) Es by (17). Since D s L : E s → E s is a self-adjoint isometry operator and we can split E s into with (D s L)| E± = ±Id E± and The orthogonal projections Π ± : E s → E ± are given by Denote by Q(z) = (D s Lz, z) Es .

Then we can write A H as
.
3. Palais-Smale condition. In this section we prove the Palais-Smale condition for A H .
Definition 3.1. Let Φ be a C 1 -functional on a Banach space E, and let c ∈ R. A sequence {x n } ∞ n=1 ⊂ E is called a Palais-Smale sequence for Φ if Φ(x n ) → c and DΦ(x n ) E * → 0. If every Palais-Smale sequence {x n } ∞ n=1 ⊂ E for Φ has a convergent subsequence then Φ is said to satisfy the Palais-Smale condition.

Proposition 2. Assume that H satisfies (H1) − (H3). Then the Palais-Smale condition is satisfied for
We prove first that {z k } ∞ k=1 is bounded. For each z k = (u k , v k ) we take It follows from (36) that there is a sequence {ε k } converging to 0 such that

WENMIN GONG AND GUANGCUN LU
By (H2) and (18) we find a constant c 1 so that Writing z ± k = (u ± k , v ± k ), we deduce from (36) and (H3) that Thus Combining (38) with (40) for z k = z + k + z − k , we obtain which implies that {z k } ∞ k=1 is bounded. Passing to a subsequence, one may assume that z k converges weakly in E s to z = (u, v). Since the operator D s L is isometric and DH is compact (see proposition (1)), we conclude that converges in E s and z k − z → 0 as k → ∞. This completes the proof.
4. Linking geometry. In this section, we give some geometric conditions for two linking properties of A H . A suitable framework for the infinite dimensional linking geometry was first given by [9]. Before giving the geometric conditions for the first linking property, we define linking subsets as follows: For R 1 , R 2 , ρ > 0 with 0 < ρ < R 2 , we define and where e + = (ξ + , η + ) ∈ E + with ξ + some eigenspinor of D corresponding to the first positive eigenvalue λ 1 . We assume e + = 1. Denote by ∂Q R1,R2 the boundary of Q R1,R2 relative to the subspace Lemma 4.1. There exist ρ > 0 and δ > 0 such that Proof. Assumptions (H1), (H3) and (H4) imply that for any ε > 0 there exists a constant C ε > 0 such that for all x ∈ M and u, v ∈ Σ x M . By (45) and the Sobolev embeddings we obtain for some constants c 1 > 0 and c 2 > 0. Thus, choosing ε = 1 4c1 , we can take ρ > 0 and δ > 0 small enough such that A H (z) ≥ δ on S ρ .
For giving the geometric conditions of the second linking property we use the following notation: Lemma 4.3. There exists ρ k > r k > 0 such that Proof. Set Then t 1 > 2 and t 2 > 2. Let z = (u, v) ∈ Z k . Using (45), by the Sobolev embeddings, we obtain where c 1 and c 2 are two constants. Choosing ε = 1 4c1 , for 0 < α k ≤ 1 and z ≥ 1, we deduce from (62) that Hence for z = r k = (4t 1 c 2 C ε α t2 k ) Similar to the proof of Lemma 3.8 in [25] we see that α k → 0 as k → ∞. Indeed, 0 < α k+1 < α k implies that α k → α ≥ 0 as k → ∞. For each each k ≥ 1 there exists z k = (u k , v k ) ∈ Z k such that z k = 1 and u k L p+1 + v k L q+1 ≥ 1 2 α k . The definition of Z k implies that z k weakly converges to 0 in E s . Then by the Sobolev imbedding theorem we obtain z k → 0 in L p+1 (M, ΣM ) × L q+1 (M, ΣM ). Therefore, by (64), relation (A 1 ) is proved.
Let z = y + w ∈ Y k with y ∈ E − and w ∈ k j=1 Re j . Assumptions (H2) and (H4) imply that for every δ > 0 there exists a constant C δ > 0 such that and then Since E + is orthogonal to E − in L 2 (M, ΣM ⊕ ΣM ), and in a finite-dimensional vector space all norms are equivalent, there exists a constant c 3 such that Combining (66) with (67) yields Taking δ > 1 2c3 , we obtain A H (z) → −∞ as z → ∞, and consequently relation (A 2 ) is satisfied.

5.
Proof of Theorem 1.1. In this section we use a variant version of Benci-Rabinowitz's generalized saddle point theorem in [9] to prove the existence of critical points of A H . Assume that E is a Hilbert space with inner product , and norm · and it has a splitting E = X ⊕ Y , where both of the subspaces X and Y can be infinite dimensional. We denote by Π X the projection of E onto X. Let Φ ∈ C 1 (E, R) be a functional having the form Let e + ∈ Y , e + = 1. For ρ, R 1 , R 2 > 0 with R 2 > ρ we set Denote by ∂Q the boundary of Q relative to the subspace {z + re + |z ∈ X, r ∈ R}.
6. Proof of Theorem 1.2. In this section we use a generalized fountain theorem to prove Theorem 1.2. Let Y be a separable closed subspace of a Hilbert space X with inner product ·, · and norm · , and Z = Y ⊥ =