Existence Results for Solutions to Nonlinear Dirac Systems on Compact Spin Manifolds

Xu Yang

doi:10.1515/ans-2017-6034

Artikel Open Access

Existence Results for Solutions to Nonlinear Dirac Systems on Compact Spin Manifolds

Xu Yang

Veröffentlicht/Copyright: 18. Oktober 2017

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Aus der Zeitschrift Advanced Nonlinear Studies Band 18 Heft 1

Abstract

In this article, we study the existence of solutions for the Dirac system

{ D ⁢ u = ∂ ⁡ H ∂ ⁡ v ⁢ ( x , u , v ) on ⁢ M , D ⁢ v = ∂ ⁡ H ∂ ⁡ u ⁢ ( x , u , v ) on ⁢ M ,

where M is an m-dimensional compact Riemannian spin manifold, u,v∈C∞⁢(M,Σ⁢M) are spinors, D is the Dirac operator on M, and the fiber preserving map H:Σ⁢M⊕Σ⁢M→ℝ is a real-valued superquadratic function of class C1 with subcritical growth rates. Two existence results of nontrivial solutions are obtained via Galerkin-type approximations and linking arguments.

Keywords: Dirac System; Compact Spin Manifolds; Variational Methods

MSC 2010: 58J05; 58E05; 53C27

1 Introduction and Main Results

Let (M,g) be an m-dimensional compact oriented Riemannian manifold equipped with a spin structure ρ:Pspin⁡(M)→Pso⁢(M), and let Σ⁢M=Pspin⁡(M)×σΣm denote the complex spinor bundle on M, 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 assume m≥2 in this paper. Write the point of Σ⁢M as (x,ϕ), where x∈M and ϕ∈Σx⁢M. The Dirac operator is an elliptic differential operator of order one,

D = D g : C ∞ ⁢ ( M , Σ ⁢ M ) → C ∞ ⁢ ( M , Σ ⁢ M ) ,

locally given by

D ⁢ ϕ = ∑ i = 1 m e i ⋅ ∇ e i ⁡ ϕ

for ϕ∈C∞⁢(M,Σ⁢M) and a local g-orthogonal frame {ei}i=1m of the tangent bundle TM. It is an unbounded essential self-adjoint operator in L2⁢(M,Σ⁢M) with the domain C∞⁢(M,Σ⁢M), and its spectrum consists of an unbounded sequence of real numbers; these mean that the closure of Dg, also denoted by Dg, is a self-adjoint operator in L2⁢(M,Σ⁢M) with the domain H1⁢(M,Σ⁢M) and has the same spectrum as the original Dg (cf. [10, 17]).

Write a point of the fiber product bundle Σ⁢M⊕Σ⁢M as (x,u,v), where x∈M and u,v∈Σx⁢M. We consider the nonlinear Dirac system

(1.1) { D ⁢ u = ∂ ⁡ H ∂ ⁡ v ⁢ ( x , u , v ) on ⁢ M , D ⁢ v = ∂ ⁡ H ∂ ⁡ u ⁢ ( x , u , v ) on ⁢ M ,

where u,v∈C1⁢(M,Σ⁢M) are spinors and the fiber preserving map H:Σ⁢M⊕Σ⁢M→ℝ is a real-valued superquadratic function of class C1 with subcritical growth rates. Problem (1.1) is the Euler–Lagrange equation of the functional

(1.2) 𝔏 ⁢ ( u , v ) = ∫ M { 〈 D ⁢ u , v 〉 - H ⁢ ( x , u , v ) } ⁢ 𝑑 x ,

where dx is the Riemann volume measure on M with respect to the metric g and 〈⋅,⋅〉 is the compatible metric on Σ⁢M.

Let

ψ = ( u , v ) , L = ( 0 D D 0 ) , L ⁢ ψ := ( D ⁢ v , D ⁢ u ) .

Then (1.2) becomes

𝔏 ⁢ ( ψ ) = ∫ M { 1 2 ⁢ 〈 L ⁢ ψ , ψ 〉 - H ⁢ ( x , ψ ) } ⁢ 𝑑 x .

Problem (1.1) can be viewed as a spinorial analogue of other strongly indefinite variational problems such as infinite dynamical systems [7, 6] and elliptic systems [4, 8]. A typical way to deal with such problems is the min-max method of Benci and Rabinowitz [20], including the mountain pass theorem, linking arguments and so on. Other approaches make use of the homological method, Morse theory and Rabinowitz–Floer homology as in [1, 5, 16]. Recently, Takeshi Isobe [13] established existence results via Galerkin-type approximations and linking arguments. Our aim in this article is to study problem (1.1) with such a method. For the nonlinear Dirac equations on a general compact spin manifold, some results were recently obtained in, e.g., [2, 3, 14, 15, 18, 19, 12, 11, 24].

For the nonlinearity H, we make the following hypotheses: H∈C0⁢(Σ⁢M⊕Σ⁢M,ℝ) is C1 in the fiber direction. Real constants 2<p,q<2∗:=2⁢mm-1 satisfy

(1.3) max ⁡ { p - 1 , q - 1 , 2 } < μ ≤ min ⁡ { p , q } ,

(1.4) ( 1 - 1 p ) ⁢ max ⁡ { p μ , q μ } < 1 2 + 1 2 ⁢ m ,

(1.5) ( 1 - 1 q ) ⁢ max ⁡ { p μ , q μ } < 1 2 + 1 2 ⁢ m ,

(1.6) p - 1 p ⁢ q μ < 1 and q - 1 q ⁢ p μ < 1 .

Moreover, we consider the following hypotheses:

There exists 0<α<1 such that H is α-Hölder continuous in the direction of the base. Moreover, there exists a constant c1>0 such that

| H u ⁢ ( x , u , v ) | ≤ c 1 ⁢ ( | u | p - 1 + | v | ( p - 1 ) ⁢ q p + 1 ) ,

| H v ⁢ ( x , u , v ) | ≤ c 1 ⁢ ( | v | q - 1 + | u | ( q - 1 ) ⁢ p q + 1 ) .
There exists R1>0 such that

0 < μ ⁢ H ⁢ ( x , u , v ) ≤ 〈 H u , u 〉 + 〈 H v , v 〉

for all (x,u,v)∈Σ⁢M⊕Σ⁢M with |(u,v)|≥R1.
H⁢(x,u,v)≥0 for all (x,u,v)∈Σ⁢M⊕Σ⁢M.
H⁢(x,u,v)=o⁢(|(u,v)|2) as |(u,v)|→0 uniformly for x∈M.
H⁢(x,-ψ)=H⁢(x,ψ) for any (x,ψ)∈Σ⁢M⊕Σ⁢M.

Notice that H⁢(x,u,v)=|u|p+|v|q satisfies these conditions.

Our main result is the following.

Theorem 1.1.

If the above H satisfies (H1)–(H4), then the Dirac system (1.1) possesses at least one solution ψ∈C1⁢(M,Σ⁢M)×C1⁢(M,Σ⁢M) with L⁢(ψ)>0.

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

Theorem 1.2.

If the above H satisfies (H1), (H2) and (H5), then there exists a sequence of solutions

{ ψ k } k = 1 ∞ ⊂ C 1 ⁢ ( M , Σ ⁢ M ) × C 1 ⁢ ( M , Σ ⁢ M )

to (1.1) with L⁢(ψk)→∞ as k→∞.

These two theorems will be proved by Galerkin-type approximations, linking arguments and the Fountain theorem.

2 Preliminaries

2.1 Spectrum of Operator L

Recall that the operator D is essentially self-adjoint in L2⁢(M,Σ⁢M) with domain 𝒟⁢(D)=C∞⁢(M,Σ⁢M) of all smooth sections of the spinor bundle on M. In particular, there exists a complete orthogonal basis {ψk} which are contained in C∞⁢(M,Σ⁢M). Similarly, we can follow the lines of [10] to prove that the operator L=(0DD0) has the same properties as D. This will be completed in several steps. Consider the product Hilbert space L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M) with inner product

(2.1) ( ψ 1 , ψ 2 ) 2 = ∫ M 〈 ψ 1 , ψ 2 〉 ⁢ 𝑑 x = ( u 1 , u 2 ) 2 + ( v 1 , v 2 ) 2

for ψi=(ui,vi)∈L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M), where

( u 1 , u 2 ) 2 = ∫ M 〈 u 1 , u 2 〉 ⁢ 𝑑 x

is the L2-inner product on spinors. Then L is a symmetric operator in L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M) with domain 𝒟⁢(L)=C∞⁢(M,Σ⁢M)×C∞⁢(M,Σ⁢M). Without occurrences of confusions we also denote by ∥⋅∥2 the norm induced by the inner product in (2.1).

Now we start to prove the essential self-adjointness of the operator L. Let L∗ be the adjoint to L. In the domain D⁢(L∗) we introduce the norm

N ⁢ ( ψ ) = ∥ ψ ∥ 2 2 + ∥ L ∗ ⁢ ψ ∥ 2 2 ,

where ∥⋅∥2 denotes the norm in the Hilbert space L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M). Denote by L¯ the closure of L. We shall prove L¯=L∗.

Lemma 2.1.

If C∞⁢(M,Σ⁢M)×C∞⁢(M,Σ⁢M)⊂D⁢(L∗) is dense with respect to the N-norm, then L is essentially self-adjoint.

Proof.

We prove the lemma in three steps. Step 1: Prove D⁢(L¯)=D⁢(D¯)×D⁢(D¯).

For ψ=(u,v)∈𝒟⁢(L¯), by the definition of L¯ we have a sequence

ψ n = ( u n , v n ) ∈ 𝒟 ⁢ ( L )

such that ∥ψn-ψ∥2→0 and ∥L⁢ψn-L¯⁢ψ∥2→0. These imply

∥ u n - u ∥ 2 → 0 , ∥ v n - v ∥ 2 → 0 , ∥ D ⁢ v n - ( L ¯ ⁢ ψ ) 1 ∥ 2 → 0 , ∥ D ⁢ u n - ( L ¯ ⁢ ψ ) 2 ∥ 2 → 0 ,

where L¯⁢ψ=((L¯⁢ψ)1,(L¯⁢ψ)2)∈L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M). These mean

u , v ∈ 𝒟 ⁢ ( D ¯ ) , D ¯ ⁢ v = ( L ¯ ⁢ ψ ) 1 , D ¯ ⁢ u = ( L ¯ ⁢ ψ ) 2 ,

and therefore 𝒟⁢(L¯)⊂𝒟⁢(D¯)×𝒟⁢(D¯) and L¯⁢ψ=(D¯⁢v,D¯⁢u) for all ψ=(u,v)∈𝒟⁢(L¯).

Conversely, let (ξ,η)∈𝒟⁢(D¯)×𝒟⁢(D¯). Then there exist sequences ξn∈𝒟⁢(D) and ηn∈𝒟⁢(D) such that

∥ ξ n - ξ ∥ 2 → 0 , ∥ D ⁢ ξ n - D ¯ ⁢ ξ ∥ 2 → 0 , ∥ η n - η ∥ 2 → 0 , ∥ D ⁢ η n - D ¯ ⁢ η ∥ 2 → 0 .

So in L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M) we have

ψ n = ( ξ n , η n ) → ψ = ( ξ , η ) , L ⁢ ψ n = ( D ⁢ η n , D ⁢ ξ n ) → ( D ¯ ⁢ η , D ¯ ⁢ ξ ) .

These mean that ψ∈𝒟⁢(L¯) and L¯ψ=(D¯v,D¯u)). That is, 𝒟⁢(D¯)×𝒟⁢(D¯)⊂𝒟⁢(L¯).

Step 2: Prove D⁢(L∗)=D⁢(D∗)×D⁢(D∗).

Let L∗ be the adjoint to L. Then 𝒟⁢(L∗) consists of those

ψ = ( u , v ) ∈ L 2 ⁢ ( M , Σ ⁢ M ) × L 2 ⁢ ( M , Σ ⁢ M )

for which there exist ϕ=(a,b)∈L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M) such that

( D ⁢ g , u ) 2 + ( D ⁢ f , v ) 2 = ( L ⁢ φ , ψ ) 2 = ( φ , ϕ ) 2 = ( f , a ) 2 + ( g , b ) 2

for all φ=(f,g)∈C∞⁢(M,Σ⁢M)×C∞⁢(M,Σ⁢M). It easily follows that 𝒟⁢(L∗)=𝒟⁢(D∗)×𝒟⁢(D∗).

Step 3. Since D is essentially self-adjoint, we have D¯=D∗, and hence 𝒟⁢(L∗)=𝒟⁢(D¯)×𝒟⁢(D¯)=𝒟⁢(L¯) by Steps 1 and 2. ∎

Since C∞⁢(M,Σ⁢M) is dense in 𝒟⁢(D∗) with respect to the N-norm (cf. [10, p. 94]) and 𝒟⁢(L∗)=𝒟⁢(D∗)×𝒟⁢(D∗), we obtain that C∞⁢(M,Σ⁢M)×C∞⁢(M,Σ⁢M) is dense in 𝒟⁢(L∗) with respect to the N-norm. So Lemma 2.1 leads to the following result.

Lemma 2.2.

Let (Mm,g) be a compact spin Riemann manifold. Then the operator L is essentially self-adjoint in L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M).

Properties of the operator D are given by the following proposition.

Proposition 2.3.

Let (Mm,g) be a compact spin Riemann manifold. Then the spectrum of L satisfies

σ p ⁢ ( L ) = σ ⁢ ( L ¯ ) , σ c ⁢ ( L ¯ ) = σ r ⁢ ( L ¯ ) = ∅ ,

and hence

σ ⁢ ( L ) = σ ⁢ ( L ¯ ) = σ p ⁢ ( L ) = σ p ⁢ ( L ¯ ) .

Proof.

Note that σ⁢(L)=σ⁢(L¯). Lemma 2.1 implies that L¯ is a self-adjoint operator, and thus L¯ has no residual spectrum, i.e., σr⁢(L¯)=∅. We have

(2.2) σ ⁢ ( L ) = σ ⁢ ( L ¯ ) = σ p ⁢ ( L ¯ ) ∪ σ c ⁢ ( L ¯ ) .

If λ is an eigenvalue of L¯, i.e., λ∈σp⁢(L¯), then there exists a spinor field ψ∈L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M) with L¯⁢(ψ)=λ⁢ψ. The regularity theorem for elliptic differential operators then implies that ψ is smooth. Hence, we have ψ∈𝒟⁢(L), i.e., the point spectrum of L coincides with the point spectrum of the closure:

σ p ⁢ ( L ) = σ p ⁢ ( L ¯ ) .

Let us prove that the residual spectrum of L is empty, i.e.,

(2.3) σ r ⁢ ( L ) = ∅ .

In fact, following the arguments in [10, p. 99], let us suppose that σr⁢(L) contains a number λ. Then there exists a spinor field 0≠φ∈L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M) such that

( L ⁢ ( ψ - λ ⁢ ψ ) , φ ) 2 = 0 for all ⁢ ψ ∈ C ∞ ⁢ ( M , Σ ⁢ M ) × C ∞ ⁢ ( M , Σ ⁢ M ) .

Choosing ψ with support in a chart and transferring this equation to the Euclidean space, we obtain an elliptic differential operator P (=L-λ) as well as a function φ∈L2⁢(ℝm,Σ⁢ℝm)×L2⁢(ℝm,Σ⁢ℝm) such that (P⁢(ψ),φ)2=0 for all ψ∈Cc∞⁢(ℝm). By the regularity theorem for elliptic operators, φ is smooth. In this case, the equation (P⁢(ψ),φ)2=0 can be written as (ψ,(L-λ)⁢φ)2=0. This in turn implies L⁢φ=λ⁢φ and φ∈𝒟⁢(L), i.e., λ∈σp⁢(L). This contradiction affirms (2.3).

From (2.2) and (2.3) we obtain

(2.4) σ p ⁢ ( L ) ∪ σ c ⁢ ( L ) = σ ⁢ ( L ) = σ ⁢ ( L ¯ ) = σ p ⁢ ( L ¯ ) ∪ σ c ⁢ ( L ¯ ) .

Recall that the approximation spectrumσα⁢(A) of an arbitrary operator A:𝒟→ℛ⁢(A) (here ℛ⁢(A) denotes the range of A) in a Hilbert space is defined by

{ λ ∈ ℂ ∣ x n ∈ 𝒟 ⁢ ( A ) ⁢ such that ⁢ ∥ x n ∥ = 1 , ∥ A ⁢ ( x n ) - λ ⁢ x n ∥ → 0 , n → ∞ } .

It is easily seen that we have always

σ p ⁢ ( A ) ∪ σ c ⁢ ( A ) ⊂ σ α ⁢ ( A ) ⊂ σ ⁢ ( A ) .

Applying this to our L gives σp⁢(L)∪σc⁢(L)⊂σα⁢(L)⊂σ⁢(L), and thus we arrive at

(2.5) σ α ⁢ ( L ) = σ α ⁢ ( L ¯ ) = σ ⁢ ( L ) = σ ⁢ ( L ¯ )

by (2.4). It remains to be shown that

(2.6) σ α ⁢ ( L ) = σ p ⁢ ( L ) .

Clearly, σp⁢(L)⊂σα⁢(L) by (2.5). Let us prove σα⁢(L)⊂σp⁢(L). Take λ∈σα⁢(L¯). Then there is a sequence of spinor fields ψn∈C∞⁢(M,Σ⁢M)×C∞⁢(M,Σ⁢M) such that

(2.7) ∥ ψ n ∥ 2 = 1 and ∥ L ⁢ ( ψ n ) - λ ⁢ ψ n ∥ 2 → 0 .

The latter implies

∥ L ⁢ ( ψ n ) - λ ⁢ ψ n ∥ 2 2 = ∥ D ⁢ v n - λ ⁢ u n ∥ 2 2 + ∥ D ⁢ u n - λ ⁢ v n ∥ 2 2 → 0 .

By [10, (∗) on p. 101] we have a constant c>0 such that

(2.8) ∥ u ∥ H 1 2 + ( min ⁡ R 4 - c - 1 ) ⁢ ∥ u ∥ 2 2 ≤ ∥ D ⁢ u ∥ 2 2 ≤ ∥ u ∥ H 1 2 + ( max ⁡ R 4 + c - 1 ) ⁢ ∥ u ∥ 2 2

for any spinor u, where R is the scalar curvature of the Riemannian manifold M. Since

1 = ∥ ψ n ∥ 2 2 = ∥ u n ∥ 2 2 + ∥ v n ∥ 2 2 ,

we deduce

∥ D ⁢ v n ∥ 2 2 = ∥ D ⁢ v n - λ ⁢ u n ∥ 2 2 + | λ | 2 ⋅ ∥ u n ∥ 2 2 + 2 ⁢ λ ⁢ ( D ⁢ v n - λ ⁢ u n , u n ) 2

≤ ∥ D ⁢ v n - λ ⁢ u n ∥ 2 2 + | λ | 2 ⋅ ∥ u n ∥ 2 2 + 2 ⁢ | λ | ⋅ ∥ u n ∥ 2 ⋅ ∥ D ⁢ v n - λ ⁢ u n ∥ 2

≤ ∥ D ⁢ v n - λ ⁢ u n ∥ 2 2 + | λ | 2 + 2 ⁢ | λ | ⋅ ∥ D ⁢ v n - λ ⁢ u n ∥ 2 ,

∥ D ⁢ u n ∥ 2 2 = ∥ D ⁢ u n - λ ⁢ v n ∥ 2 2 + | λ | 2 ⋅ ∥ v n ∥ 2 2 + 2 ⁢ λ ⁢ ( D ⁢ u n - λ ⁢ v n , v n ) 2

≤ ∥ D ⁢ u n - λ ⁢ v n ∥ 2 2 + | λ | 2 + 2 ⁢ | λ | ⋅ ∥ D ⁢ u n - λ ⁢ v n ∥ 2 .

These and (2.8) lead to

∥ u n ∥ H 1 2 + ∥ v n ∥ H 1 2 ≤ ∥ D ⁢ u n ∥ 2 2 + ∥ D ⁢ v n ∥ 2 2 + | min ⁡ R 4 - c - 1 | ⋅ ( ∥ u n ∥ 2 2 + ∥ v n ∥ 2 2 )

≤ ∥ D ⁢ u n - λ ⁢ v n ∥ 2 2 + 2 ⁢ | λ | ⋅ ∥ D ⁢ u n - λ ⁢ v n ∥ 2 + ∥ D ⁢ v n - λ ⁢ u n ∥ 2 2

+ 2 ⁢ | λ | ⋅ ∥ D ⁢ v n - λ ⁢ u n ∥ 2 + | min ⁡ R 4 - c - 1 | + 2 ⁢ | λ | 2 ,

and therefore ψn=(un,vn) is a bounded sequence in the Sobolev space H1⁢(M,Σ⁢M)×H1⁢(M,Σ⁢M) by (2.7). Moreover, the embedding H1⁢(M,Σ⁢M)×H1⁢(M,Σ⁢M)↪L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M) is compact by the Rellich lemma. Passing to a subsequence, if necessary, we can assume that ψn converges to a spinor field ψ0 in L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M), and this immediately implies L⁢ψ0=λ⁢ψ0. Note that ∥ψ0∥2=limn→∞⁡∥ψn∥2=1 by (2.7). It follows that λ is an eigenvalue of L, and hence that σα⁢(L)⊂σp⁢(L). Equation (2.6) is proved. ∎

Since ∥L⁢(ψn)∥22=∥D⁢un∥22+∥D⁢vn∥22 and the closure D¯=D∗ of the Dirac operator D is defined on the subspace H1⁢(M,Σ⁢M)⊂L2⁢(M,Σ⁢M), we immediately obtain the following proposition.

Proposition 2.4.

Under the above notation, the closure L¯=L∗ of L is defined on the subspace

H 1 ⁢ ( M , Σ ⁢ M ) × H 1 ⁢ ( M , Σ ⁢ M ) ⊂ L 2 ⁢ ( M , Σ ⁢ M ) × L 2 ⁢ ( M , Σ ⁢ M ) .

Proposition 2.5.

There exists a complete orthonormal basis ψ1,ψ2,… of the Hilbert space of

L 2 ⁢ ( M , Σ ⁢ M ) × L 2 ⁢ ( M , Σ ⁢ M )

consisting of eigenspinors of the operator L, which are contained in C∞⁢(M,Σ⁢M)×C∞⁢(M,Σ⁢M), i.e., L⁢(ψn)=λ¯⁢ψn. Moreover, limn→∞⁡|λ¯n|=∞ and all corresponding eigenvalues have finite multiplicity.

Proof.

Let λ¯∈σ⁢(L). Since σ⁢(L)=σp⁢(L) by Proposition 2.3, there is a nonzero

ψ = ( u , v ) ∈ L 2 ⁢ ( M , Σ ⁢ M ) × L 2 ⁢ ( M , Σ ⁢ M )

such that L⁢(u,v)=λ¯⁢(u,v), which implies that D⁢u=λ¯⁢v and D⁢v=λ¯⁢u. So

D ⁢ ( u + v ) = λ ¯ ⁢ ( u + v ) and D ⁢ ( u - v ) = - λ ¯ ⁢ ( u - v ) .

Note that either u+v≠0 or u-v≠0. We get that either λ¯∈σ⁢(D) or -λ¯∈σ⁢(D). That is, σ⁢(L)⊂σ⁢(D)∪(-σ⁢(D)). Then the conclusion follows from [10, proposition on p. 102]. Of course, we can also deduce the claim as in the proof of [10, second proposition on p. 101]. In fact, by (2.8) we have

∥ ψ ∥ H 1 2 + ( min ⁡ R 4 - c - 1 ) ⁢ ∥ ψ ∥ 2 2 = ∥ u ∥ H 1 2 + ( min ⁡ R 4 - c - 1 ) ⁢ ∥ u ∥ 2 2 + ∥ v ∥ H 1 2 + ( min ⁡ R 4 - c - 1 ) ⁢ ∥ v ∥ 2 2

≤ ∥ D ⁢ u ∥ 2 2 + ∥ D ⁢ v ∥ 2 2 = ∥ L ⁢ ψ ∥ 2 2

for any ψ=(u,v)∈H1⁢(M,Σ⁢M)×H1⁢(M,Σ⁢M). As in the arguments in [10, p. 101], we derive from this that for each λ∈ℝ∖σ⁢(L¯) the operator

( L - λ ) - 1 : L 2 ⁢ ( M , Σ ⁢ M ) × L 2 ⁢ ( M , Σ ⁢ M ) → L 2 ⁢ ( M , Σ ⁢ M ) × L 2 ⁢ ( M , Σ ⁢ M )

is compact. Then the conclusions can be derived as in [10, p. 102]. ∎

2.2 Sobolev Spaces

Recall that the closure of the operator D, still denoted by D, is an unbounded self-adjoint operator in L2⁢(M,Σ⁢M) with domain H1⁢(M,Σ⁢M). Moreover, there exists a complete orthonormal basis η1,η2,… of the Hilbert space L2⁢(M,Σ⁢M) which are contained in C∞⁢(M,Σ⁢M) consisting of the eigenspinors of the operator D, i.e. D⁢ηk=λk⁢ηk. Moreover, |λk|→∞ (as k→∞) and all corresponding eigenvalues λ1,λ2,… have finite multiplicity.

For s≥0, we define the unbounded operator |D|s:L2⁢(M,Σ⁢M)→L2⁢(M,Σ⁢M) by

| D | s ⁢ η = ∑ k = 1 ∞ | λ k | s ⁢ a k ⁢ η k ,

where η=∑k=1∞ak⁢ηk∈L2⁢(M,Σ⁢M). Denote by Hs⁢(M,Σ⁢M) the domain of it. Then

η = ∑ k = 1 ∞ a k ⁢ η k ∈ H s ⁢ ( M , Σ ⁢ M )

if and only if

∑ k = 1 ∞ | λ k | 2 ⁢ s ⁢ | a k | 2 < ∞ .

Note that Hs⁢(M,Σ⁢M) coincides with the usual L2-Sobolev space of order s, Ws,2⁢(M,Σ⁢M). The inner product on Hs⁢(M,Σ⁢M) is defined by

( η , ζ ) s , 2 := ( | D | s ⁢ η , | D | s ⁢ ζ ) 2 + ( η , ζ ) 2 ,

and the induced norm is denoted by ∥⋅∥s,2, i.e.,

∥ η ∥ s , 2 = ( η , η ) s , 2 1 2 .

As usual, we write

∥ η ∥ p = ( ∫ M | η | p ) 1 p

for the Lp-norm of η∈Lp⁢(M,Σ⁢M). The Sobolev space H1/2⁢(M,Σ⁢M) is the largest Sobolev space where the integral ∫M〈η,D⁢η〉⁢𝑑x is well defined. By the Sobolev embedding theorem, we have the continuous embedding H1/2⁢(M,Σ⁢M)↪Lp⁢(M,Σ⁢M) for 1≤p≤2∗. Moreover, it is also compact if 1≤p<2∗.

To treat the Dirac system (1.1) from a variational point of view, it is necessary to give a suitable functional analytic framework. A suitable function space to work with the functional 𝔏 is the Sobolev space E1/2:=H1/2⁢(M,Σ⁢M)×H1/2⁢(M,Σ⁢M) with norm

∥ ψ ∥ E 1 / 2 = ( ∥ u ∥ 1 / 2 , 2 2 + ∥ v ∥ 1 / 2 , 2 2 ) 1 2

for ψ=(u,v)∈E1/2.

By Proposition 2.5, the closure of the operator L, still denoted by L, is an unbounded self-adjoint operator in L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M) with domain H1⁢(M,Σ⁢M)×H1⁢(M,Σ⁢M). Moreover, the spectrum of L consists of eigenvalues satisfying

- ∞ ↓ λ ¯ k - ≤ ⋯ ≤ λ ¯ 1 - < 0 < λ ¯ 1 + ≤ ⋯ ≤ λ ¯ k + ↑ + ∞ , k ∈ ℕ .

The complete orthonormal basis {ψk} of L2⁢(M,Σ⁢M)×L2⁢(M,Σ⁢M) consisting of the eigenspinors of L is decomposed into three parts:

{ ψ k } k = 1 ∞ = { ψ k - } k = 1 ∞ ∪ { ψ k 0 } k = 1 l ∪ { ψ k + } k = 1 ∞ ,

where L⁢ψk-=λ¯k-⁢ψk-, L⁢ψk+=λ¯k+⁢ψk+ and L⁢ψk0=0, k=1,…,l=dim⁡ker⁡(L)<∞. By the elliptic regularity,

{ ψ k } k = 1 ∞ ⊂ C ∞ ⁢ ( M , Σ ⁢ M ) × C ∞ ⁢ ( M , Σ ⁢ M ) .

Set

E - := span { ψ k - } k = 1 ∞ ¯ , E 0 := span { ψ k 0 } k = 1 l , E + := span { ψ k + } k = 1 ∞ ¯ ,

where the closure is taken in the E1/2-topology. We then have the orthogonal decomposition of the Hilbert space:

E 1 2 = E - ⊕ E 0 ⊕ E + .

Under the growth hypothesis (H1), using Young’s inequality, we derive

(2.9) | H ⁢ ( x , u , v ) | ≤ c 2 ⁢ ( | u | p + | v | q ) + c 3 ,

which implies that the functional 𝔏 is well defined on E1/2. By the compactness of the inclusion

E 1 2 ↪ L p ⁢ ( M , Σ ⁢ M ) × L q ⁢ ( M , Σ ⁢ M ) ,

we can define the functional ℋ:E1/2→ℝ by

ℋ ⁢ ( u , v ) = ∫ M H ⁢ ( x , u , v ) ⁢ 𝑑 x .

Proposition 2.6 ([11, Proposition 2.1]).

Assume that H satisfies condition (H1). Then the functional H is of class C1 and its derivative is given by

ℋ ′ ⁢ ( u , v ) ⁢ ( h 1 , h 2 ) = ∫ M { 〈 H u , h 1 〉 + 〈 H v , h 2 〉 } ⁢ 𝑑 x = ∫ M 〈 H ψ , h 〉 ⁢ 𝑑 x

for all ψ=(u,v),h=(h1,h2)∈E1/2. Moreover, H′:E1/2→E12* is a compact operator.

It follows that for ψ=(u,v),h=(h1,h2)∈E1/2 we have

〈 d ⁢ 𝔏 ⁢ ( ψ ) , h 〉 = ∫ M 〈 D ⁢ v , h 1 〉 + 〈 D ⁢ u , h 2 〉 ⁢ d ⁢ x - ∫ M 〈 H u , h 1 〉 + 〈 H v , h 2 〉 ⁢ d ⁢ x

= ∫ M 〈 L ⁢ ψ , h 〉 ⁢ 𝑑 x - ∫ M 〈 H ψ , h 〉 ⁢ 𝑑 x .

Thus the critical points of 𝔏 correspond to solutions of the Dirac system (1.1) as asserted.

3 Palais–Smale Condition for 𝔏n

Let F be a C1 functional on a Banach space H. Recall that a sequence {xn}⊂H is called a (PS)c-sequence if F⁢(xn)→c as n→∞ and ∥d⁢F⁢(xn)∥H*→0 as n→∞. If all (PS)c -sequences converge in H, we say that F satisfies the (PS)c condition. As in [13], the following modified version of the condition is needed.

Definition 3.1.

Let F:H→ℝ be as above, and let {H⁢(n)}n=1∞ be a sequence of closed subspaces in H satisfying H⁢(n)⊂H⁢(n+1) for all n∈ℕ. Set Fn:=F∣H⁢(n), the restriction of F to H⁢(n). A sequence {xn}⊂H is called a (PS)c*-sequence with respect to {H⁢(n)} if xn∈H⁢(n) for all n and

F⁢(xn)→c as n→∞,
∥d⁢Fn⁢(xn)∥H⁢(n)*→0 as n→∞.

If all (PS)c* sequences converge in H, we say that F satisfies the (PS)c* condition with respect to {H⁢(n)}.

Write E11/2=E-⊕E0 and E21/2=E+. Then E1/2=E11/2⊕E21/2. For n≥l=dim⁡ker⁡(L), we define

E 2 1 2 ⁢ ( n ) := span ⁡ ( { ψ k - } k = 1 n - l ∪ { ψ k 0 } k = 1 l ) ⊕ E 2 1 2

equipped with the E1/2-norm.

Lemma 3.2.

Suppose H satisfies (H1) and (H2). Then for any c∈R the functional L satisfies the (PS)c*-condition with respect to {E21/2⁢(n)}.

Proof.

Write

𝔏 n = 𝔏 | E 2 1 / 2 ⁢ ( n ) ,

and let {ψn}={(un,vn)}⊂E1/2 be a (PS)c*-sequence with respect to {E21/2⁢(n)}, i.e., ψn∈E21/2⁢(n) and it satisfies

(3.1) 𝔏 ⁢ ( ψ n ) → c as ⁢ n → ∞

and

(3.2) ∥ d ⁢ 𝔏 n ⁢ ( ψ n ) ∥ E 2 1 / 2 ⁢ ( n ) * → 0 as ⁢ n → ∞ .

Claim 1: {ψn}⊂E1/2 is bounded.

Condition (H2) implies that there are constants c4,c5>0 such that

(3.3) H ⁢ ( x , u , v ) ≥ c 4 ⁢ ( | u n | μ + | v n | μ ) - c 5 .

(see [9] for a proof). By (3.1)–(3.3), for large n we have

C + ∥ ψ n ∥ E 1 / 2 ≥ 2 ⁢ 𝔏 ⁢ ( ψ n ) - 〈 d ⁢ 𝔏 n ⁢ ( ψ n ) , ψ n 〉

= ∫ M 〈 H ψ , ψ n 〉 ⁢ 𝑑 x - 2 ⁢ ∫ M H ⁢ ( x , ψ n ) ⁢ 𝑑 x

≥ ( μ - 2 ) ⁢ ∫ M H ⁢ ( x , ψ n ) ⁢ 𝑑 x - C

(3.4) ≥ c 4 ⁢ ( μ - 2 ) ⁢ ∫ M | u n | μ + | v n | μ ⁢ d ⁢ x - c 5 ⁢ ( μ - 2 ) - C .

Hereafter, C denotes various positive constants which do not depend on n. Clearly, (3.4) implies that

∥ u n ∥ μ μ + ∥ v n ∥ μ μ ≤ C ⁢ ( 1 + ∥ ψ n ∥ E 1 / 2 )

for large n, and so

(3.5) ∥ u n ∥ μ μ ≤ C ⁢ ( 1 + ∥ ψ n ∥ E 1 / 2 ) and ∥ v n ∥ μ μ ≤ C ⁢ ( 1 + ∥ ψ n ∥ E 1 / 2 ) .

Write ψn=ψn-+ψn0+ψn+ according to the decomposition E1/2=E-⊕E0⊕E+. Then we have

∥ ψ + ∥ E 1 / 2 2 = ∥ ( u + , v + ) ∥ E 1 / 2 2

= ∫ M ( | | D | 1 2 ⁢ u + | 2 + | | D | 1 2 ⁢ v + | 2 + | u + | 2 + | v + | 2 ) ⁢ 𝑑 x

= ∫ M 〈 D ⁢ u + , u + 〉 + 〈 D ⁢ v + , v + 〉 ⁢ d ⁢ x + ∫ M ( | u + | 2 + | v + | 2 ) ⁢ 𝑑 x

≤ ∫ M 〈 L ⁢ ψ + , ψ + 〉 ⁢ 𝑑 x + 1 λ 1 + ⁢ ∫ M 〈 L ⁢ ψ + , ψ + 〉 ⁢ 𝑑 x

= ( 1 + 1 λ 1 + ) ⁢ ∫ M 〈 L ⁢ ψ + , ψ + 〉 ⁢ 𝑑 x .

Set

C + = 1 2 ⁢ ( 1 + 1 λ 1 + ) - 1 .

Then

(3.6) C + ⁢ ∥ ψ + ∥ E 1 / 2 2 ≤ 1 2 ⁢ ∫ M 〈 L ⁢ ψ + , ψ + 〉 ⁢ 𝑑 x .

By (3.2), for large n we have

| 〈 d ⁢ 𝔏 n ⁢ ( ψ n ) , ψ n + 〉 | = | ∫ M 〈 ψ n + , L ⁢ ψ n 〉 ⁢ 𝑑 x - ∫ M 〈 ψ n + , H ψ 〉 ⁢ 𝑑 x | ≤ ∥ ψ n + ∥ E 1 / 2 .

This and (H1) lead to

| ∫ M 〈 ψ n + , L ⁢ ψ n 〉 ⁢ 𝑑 x | ≤ | ∫ M 〈 ψ n + , H ψ 〉 ⁢ 𝑑 x | + ∥ ψ n + ∥ E 1 / 2

≤ | ∫ M 〈 u n + , H u 〉 ⁢ 𝑑 x | + | ∫ M 〈 v n + , H v 〉 ⁢ 𝑑 x | + ∥ ψ n + ∥ E 1 / 2

≤ ∫ M | u n + | ⁢ | H u | ⁢ 𝑑 x + ∫ M | v n + | ⁢ | H v | ⁢ 𝑑 x + ∥ ψ n + ∥ E 1 / 2

(3.7) ≤ ∫ M c 1 ⁢ ( | u n | p - 1 + | v n | ( p - 1 ) ⁢ q p + 1 ) ⁢ | u n + | ⁢ 𝑑 x + ∫ M c 1 ⁢ ( | v n | q - 1 + | u n | ( q - 1 ) ⁢ p q + 1 ) ⁢ | v n + | + ∥ ψ n + ∥ E 1 / 2 .

Note that

1 < μ μ - p + 1 < 2 ∗ and 1 < μ ⁢ p μ ⁢ p - ( p - 1 ) ⁢ q < 2 ∗

by conditions (1.4) and (1.6). We derive

H 1 2 ⁢ ( M , Σ ⁢ M ) ↪ L μ μ - p + 1 , H 1 2 ⁢ ( M , Σ ⁢ M ) ↪ L μ ⁢ p μ ⁢ p - ( p - 1 ) ⁢ q ,

and therefore

(3.8) ∫ M | u n | p - 1 ⁢ | u n + | ⁢ 𝑑 x ≤ C ⁢ ∥ u n ∥ μ p - 1 ⁢ ∥ u n + ∥ 1 2 , 2 ,

(3.9) ∫ M | v n | ( p - 1 ) ⁢ q p ⁢ | u n + | ⁢ 𝑑 x ≤ C ⁢ ∥ v n ∥ μ ( p - 1 ) ⁢ q p ⁢ ∥ u n + ∥ 1 2 , 2 .

By using conditions (1.5) and (1.6), an analogous reasoning yields

(3.10) ∫ M | v n | q - 1 ⁢ | v n + | ⁢ 𝑑 x ≤ C ⁢ ∥ v n ∥ μ q - 1 ⁢ ∥ v n + ∥ 1 2 , 2 ,

(3.11) ∫ M | u n | ( q - 1 ) ⁢ p q ⁢ | v n + | ⁢ 𝑑 x ≤ C ⁢ ∥ u n ∥ μ ( q - 1 ) ⁢ p q ⁢ ∥ v n + ∥ 1 2 , 2 .

Moreover, it also holds that

(3.12) ∫ M | u n + | ⁢ 𝑑 x ≤ C ⁢ ∥ u n + ∥ 1 2 , 2 , ∫ M | v n + | ⁢ 𝑑 x ≤ C ⁢ ∥ v n + ∥ 1 2 , 2 .

By (3.6) and (3.7)–(3.12), we deduce

∥ ψ n + ∥ E 1 / 2 2 ≤ C ⁢ ∫ M 〈 L ⁢ ψ n + , ψ n + 〉 ⁢ 𝑑 x

= C ⁢ | ∫ M 〈 L ⁢ ψ n + , ψ n 〉 ⁢ 𝑑 x |

≤ ∥ ψ n + ∥ E 1 / 2 + C ⁢ ( ∥ u n ∥ μ p - 1 + ∥ v n ∥ μ ( p - 1 ) ⁢ q p + 1 ) ⁢ ∥ u n + ∥ 1 2 , 2 + C ⁢ ( ∥ v n ∥ μ q - 1 + ∥ u n ∥ μ ( q - 1 ) ⁢ p q + 1 ) ⁢ ∥ v n + ∥ 1 2 , 2 .

Hence, this and (3.5) lead to

(3.13) ∥ ψ n + ∥ E 1 / 2 2 ≤ C ⁢ ( ∥ ψ n ∥ E 1 / 2 p - 1 μ + ∥ ψ n ∥ E 1 / 2 ( p - 1 ) ⁢ q μ ⁢ p + 1 ) ⁢ ∥ ψ n ∥ E 1 / 2 + C ⁢ ( ∥ ψ n ∥ E 1 / 2 q - 1 μ + ∥ ψ n ∥ E 1 / 2 ( q - 1 ) ⁢ p μ ⁢ q + 1 ) ⁢ ∥ ψ n ∥ E 1 / 2 + ∥ ψ n ∥ E 1 / 2 .

As in (3.6), for any ψ-∈E- we also have

(3.14) 1 2 ⁢ ∫ M 〈 ψ - , L ⁢ ψ - 〉 ⁢ 𝑑 x ≤ - C - ⁢ ∥ ψ - ∥ E 1 / 2 2 ,

where C-=12⁢(1-(λ1-)-1)-1>0. Then similar arguments lead to

(3.15) ∥ ψ n - ∥ E 1 / 2 2 ≤ C ⁢ ( ∥ ψ n ∥ E 1 / 2 p - 1 μ + ∥ ψ n ∥ E 1 / 2 ( p - 1 ) ⁢ q μ ⁢ p + 1 ) ⁢ ∥ ψ n ∥ E 1 / 2 + ( ∥ ψ n ∥ E 1 / 2 q - 1 μ + ∥ ψ n ∥ E 1 / 2 ( q - 1 ) ⁢ p μ ⁢ q + 1 ) ⁢ ∥ ψ n ∥ E 1 / 2 + ∥ ψ n ∥ E 1 / 2 .

Note that all norms on the finite dimension space E0 are equivalent. We obtain

(3.16) ∥ ψ n 0 ∥ E 1 / 2 2 ≤ C ⁢ ∥ ψ n 0 ∥ 2 2 ≤ C ⁢ ∥ ψ n ∥ μ 2 ≤ C ⁢ ( 1 + ∥ ψ n ∥ E 1 / 2 2 μ ) .

Adding (3.13), (3.15) and (3.16) yields

∥ ψ n ∥ E 1 / 2 2 ≤ C ⁢ ( ∥ ψ n ∥ E 1 / 2 p - 1 μ + ∥ ψ n ∥ E 1 / 2 ( p - 1 ) ⁢ q μ ⁢ p + 1 ) ⁢ ∥ ψ n ∥ E 1 / 2 + C ⁢ ( ∥ ψ n ∥ E 1 / 2 q - 1 μ + ∥ ψ n ∥ E 1 / 2 ( q - 1 ) ⁢ p μ ⁢ q + 1 ) ⁢ ∥ ψ n ∥ E 1 / 2

(3.17) + 2 ⁢ ∥ ψ n ∥ E 1 / 2 + C ⁢ ∥ ψ n ∥ E 1 / 2 2 μ + C .

By the assumptions on p, q and μ above (H1), it is easily checked that the total exponent of each term in the right-hand side of (3) is less than 2. It follows that the sequence {ψn} is bounded in E1/2. Claim 1 is proved.

Claim 2: {ψn}={(un,vn)} has a convergent subsequence in E1/2.

Passing to a subsequence, we may assume that for some ψ=(u,v)∈E1/2,

ψ n ⇀ ψ weakly in ⁢ E 1 2 ,

ψ n → ψ strongly in ⁢ L p ⁢ ( M , Σ ⁢ M ) × L p ⁢ ( M , Σ ⁢ M ) .

From (3.2) and the boundedness of {ψn} in E1/2 we deduce

o ⁢ ( 1 ) = 〈 d ⁢ 𝔏 n ⁢ ( ψ n ) , ψ n + - ψ + 〉

= ∫ M 〈 L ⁢ ( ψ n ) , ψ n + - ψ + 〉 ⁢ 𝑑 x - ∫ M 〈 H ψ ⁢ ( x , ψ n ) , ψ n + - ψ + 〉 ⁢ 𝑑 x

(3.18) = ∫ M 〈 L ⁢ ( ψ n ) , ψ n + - ψ + 〉 ⁢ 𝑑 x - ∫ M 〈 H u ⁢ ( x , ψ n ) , u n + - u + 〉 ⁢ 𝑑 x - ∫ M 〈 H v ⁢ ( x , ψ n ) , v n + - v + 〉 ⁢ 𝑑 x .

It follows from (H1) that

| ∫ M 〈 H u ⁢ ( x , ψ n ) , u n + - u + 〉 ⁢ 𝑑 x | ≤ ∫ M | H u ⁢ ( x , ψ n ) | ⁢ | u n + - u + | ⁢ 𝑑 x

≤ ∫ M c 1 ⁢ ( | u n | p - 1 + | v n | ( p - 1 ) ⁢ q p + 1 ) ⁢ | u n + - u + | ⁢ 𝑑 x

≤ c 1 ⁢ ( ∥ u n + - u + ∥ 1 + ∥ u n ∥ p ( p - 1 ) ⁢ ∥ u n + - u + ∥ p + ∥ v n ∥ μ ( p - 1 ) ⁢ q p ⁢ ∥ u n + - u + ∥ μ ⁢ p μ ⁢ p - ( p - 1 ) ⁢ q )

(3.19) = o ⁢ ( 1 ) as ⁢ n → ∞ .

Similarly, we can also arrive at

(3.20) | ∫ M 〈 H v ⁢ ( x , ψ n ) , v n + - v + 〉 ⁢ 𝑑 x | = o ⁢ ( 1 ) as ⁢ n → ∞ .

Then (3.18)–(3.20) yield

| ∫ M 〈 L ⁢ ( ψ n ) , ψ n + - ψ + 〉 ⁢ 𝑑 x | → 0 as ⁢ n → ∞ ,

and thus

(3.21) ∥ ψ n + - ψ + ∥ E 1 / 2 2 ≤ C + - 1 2 ⁢ ∫ M 〈 L ⁢ ( ψ n + - ψ + ) , ψ n + - ψ + 〉 ⁢ 𝑑 x = o ⁢ ( 1 ) as ⁢ n → ∞

by (3.6). Note that dim⁡E0=l<∞ implies

(3.22) ∥ ψ n 0 - ψ 0 ∥ E 1 / 2 → 0 as ⁢ n → ∞ .

In order to prove ψn-→ψ- let Pn:E1/2→E21/2⁢(n) denote the orthogonal projection. Note that Pn⁢φ→φ in E1/2 for any φ∈E1/2. Hence,

o ⁢ ( 1 ) = 〈 d ⁢ 𝔏 n ⁢ ( ψ n ) , ψ n + - P n ⁢ ψ + 〉

(3.23) = ∫ M 〈 L ⁢ ( ψ n ) , ψ n + - P n ⁢ ψ + 〉 ⁢ 𝑑 x - ∫ M 〈 H ψ ⁢ ( x , ψ n ) , ψ n + - P n ⁢ ψ + 〉 ⁢ 𝑑 x .

As in (3.19) and (3.20), we can show that the second term of (3.23) converges to 0 as n→∞. Moreover, by (3.21) and (3.22) we obtain

o ⁢ ( 1 ) = ∫ M 〈 L ⁢ ( ψ n ) , ψ n - P n ⁢ ψ 〉 ⁢ 𝑑 x = ∫ M 〈 L ⁢ ( ψ n - ) , ψ n - - P n ⁢ ψ - 〉 ⁢ 𝑑 x + o ⁢ ( 1 ) ,

and therefore

o ⁢ ( 1 ) = ∫ M 〈 L ⁢ ( ψ n ) , ψ n - - P n ⁢ ψ - 〉 ⁢ 𝑑 x = ∫ M 〈 L ⁢ ( ψ n - ) , ψ n - - ψ - 〉 ⁢ 𝑑 x + o ⁢ ( 1 ) .

These and (3.14) lead to

(3.24) ∥ ψ n - - ψ - ∥ E 1 / 2 2 ≤ - C - - 1 2 ⁢ ∫ M 〈 L ⁢ ( ψ n - - ψ - ) , ψ n - - ψ - 〉 ⁢ 𝑑 x = o ⁢ ( 1 )

as n→∞. Combing (3.21) with (3.22) and (3.24), we deduce that ψn→ψ in E1/2. This proves Claim 2, and so the (PS)c∗-condition is verified. ∎

Remark 3.3.

By essentially the same argument, it can be shown that for any n and c∈ℝ, the functional 𝔏n satisfies the (PS)c-condition on E21/2⁢(n).

4 Proofs of the Main Results

4.1 Proof of Theorem 1.1

Taking e+∈E+ such that ∥e+∥E1/2=1, we define for r>0, R>0 and ρ>0,

Q r , R := { ψ - + ψ 0 + s ⁢ e + ∣ ψ - ∈ E - , ψ 0 ∈ E 0 , 0 ≤ s ≤ r , ∥ ψ - + ψ 0 ∥ E 1 / 2 ≤ R }

and

S ρ + := { ψ ∈ E + ∣ ∥ ψ ∥ E 1 / 2 = ρ } .

We need to give a uniform estimate for approximate critical levels of 𝔏. To this end, let us estimate the quantities

α r , R := sup ⁡ { 𝔏 ⁢ ( ψ ) ∣ ψ ∈ ∂ ⁡ Q r , R }

and

β ρ := inf ⁡ { 𝔏 ⁢ ( ψ ) ∣ ψ ∈ S ρ + } .

Lemma 4.1.

Suppose that H satisfies (H2) and (H3). Then there exist r>0 and R>0 such that αr,R≤0.

Proof.

For ψ=(u,v)=ψ-+ψ0+s⁢e+∈∂⁡Qr,R, a direct computation gives

𝔏 ⁢ ( ψ ) = 1 2 ⁢ ∫ M 〈 ψ - , L ⁢ ψ - 〉 ⁢ 𝑑 x + s 2 2 ⁢ ∫ M 〈 e + , L ⁢ e + 〉 ⁢ 𝑑 x - ∫ M H ⁢ ( x , ψ - + ψ 0 + s ⁢ e + ) ⁢ 𝑑 x .

Set

C e + := 1 2 ⁢ ∫ M 〈 e + , L ⁢ e + 〉 ⁢ 𝑑 x > 0 .

We can derive from (3.3) that

(4.1) ∫ M H ⁢ ( x , ψ ) ⁢ 𝑑 x ≥ c 4 ⁢ ∫ M | u | μ + | v | μ ⁢ d ⁢ x - c 5 .

Moreover, by Hölder’s inequality we have

∫ M | ψ - | 2 ⁢ 𝑑 x + ∫ M | ψ 0 | 2 ⁢ 𝑑 x + s 2 ⁢ ∫ M | e + | 2 ⁢ 𝑑 x = ∫ M | ψ | 2 ⁢ 𝑑 x

= ∫ M | u | 2 + | v | 2 ⁢ d ⁢ x

≤ ( vol ( M ) 1 - 2 μ ) ( ∫ M | u | μ d x ) 2 μ + ( vol ( M ) 1 - 2 μ ) ( ∫ M | v | μ d x ) 2 μ .

This and (4.1) lead to

(4.2) 𝔏 ⁢ ( ψ ) ≤ 1 2 ⁢ ∫ M 〈 ψ - , L ⁢ ψ - 〉 ⁢ 𝑑 x + C e + ⁢ s 2 - C ⁢ ( ∥ ψ - ∥ 2 2 + ∥ ψ 0 ∥ 2 2 + s 2 ⁢ ∥ e + ∥ 2 2 ) μ 2 + c 5 .

Since all norms on the finite dimension space E0 are equivalent, it follows from (4.2) and (3.14) that

𝔏 ⁢ ( ψ ) ≤ - C - ⁢ ∥ ψ - ∥ E 1 / 2 2 + C e + ⁢ s 2 - C ⁢ ( ∥ ψ - ∥ 2 2 + ∥ ψ 0 ∥ 2 2 + s 2 ⁢ ∥ e + ∥ 2 2 ) μ 2 + c 5

(4.3) ≤ - C - ⁢ ∥ ψ - ∥ E 1 / 2 2 + C e + ⁢ s 2 - C ⁢ ( ∥ ψ 0 ∥ E 1 / 2 μ + s μ ) + c 5 .

Note that μ>2. We can choose r>0 so that Ce+⁢s2-C⁢sμ+c5≤0 for s≥r. Set

M := max 0 ≤ s ≤ r ⁡ ( C e + ⁢ s 2 - C ⁢ s μ + c 5 ) > 0 .

Then we can choose R>0 such that

∥ ψ - + ψ 0 ∥ E 1 / 2 ≥ R ⟹ C - ⁢ ∥ ψ - ∥ E 1 / 2 2 + C ⁢ ∥ ψ 0 ∥ E 1 / 2 μ ≥ M .

There are three cases:

0≤s≤r and ∥ψ-+ψ0∥E1/2=R;
s=0 and ∥ψ-+ψ0∥E1/2≤R;
s=r and ∥ψ-+ψ0∥E1/2≤R.

For the first case, our arguments above have showed that 𝔏⁢(ψ)≤0. In the second case, it follows from (H3) that

𝔏 ⁢ ( ψ ) = 1 2 ⁢ ∫ M 〈 ψ - , L ⁢ ψ - 〉 ⁢ 𝑑 x - ∫ M H ⁢ ( x , ψ ) ⁢ 𝑑 x ≤ 0 .

Inequality (4.3) and the choices of r and R lead to 𝔏⁢(ψ)≤0 in the final case.

Summing up the three cases, we complete the proof. ∎

Lemma 4.2.

Assume that H satisfies (H1), (H3) and (H4). Then for r>0 in Lemma 4.1, there exists 0<ρ<r such that βρ>0.

Proof.

By (2.9), (H3) and (H4), there exists c6>0 such that

(4.4) 0 ≤ H ⁢ ( x , u , v ) ≤ C + 2 ⁢ ( | u | 2 + | v | 2 ) + c 6 ⁢ ( | u | p + | v | q )

for any (x,u,v)∈Σ⁢M⊕Σ⁢M. It follows from this, (3.5) and (4.4) that

𝔏 ⁢ ( ψ ) = 1 2 ⁢ ∫ M 〈 ψ , L ⁢ ψ 〉 ⁢ 𝑑 x - ∫ M H ⁢ ( x , ψ ) ⁢ 𝑑 x

≥ C + 2 ⁢ ∥ ψ ∥ E 1 / 2 2 - c 6 ⁢ ∫ M ( | u | p + | v | q ) ⁢ 𝑑 x

≥ C + 2 ⁢ ( ∥ u ∥ 1 2 , 2 2 + ∥ v ∥ 1 2 , 2 2 ) - c 7 ⁢ ( ∥ u ∥ 1 2 , 2 p + ∥ v ∥ 1 2 , 2 q )

= C + 2 ⁢ ( ρ 1 2 + ρ 2 2 ) - c 7 ⁢ ( ρ 1 p + ρ 2 q )

for ψ=(u,v)∈Sρ+, where

ρ 1 = ∥ u ∥ 1 2 , 2 , ρ 2 = ∥ v ∥ 1 2 , 2 ,

and thus they satisfy ρ2=ρ12+ρ22 because of

∥ ψ ∥ E 1 / 2 2 = ∥ u ∥ 1 2 , 2 2 + ∥ v ∥ 1 2 , 2 2 .

Note that p>2 and q>2. We can choose ρ∈(0,r) such that

C + 2 ⁢ ( ρ 1 2 + ρ 2 2 ) - c 7 ⁢ ( ρ 1 p + ρ 2 q ) > 0 .

It follows that the conclusion of the lemma holds for this ρ. ∎

Completing the proof of Theorem 1.1.

Let r>0 and R>0 be as in Lemma 4.1. Then Qr,R⁢(n):=Qr,R∩E21/2⁢(n) is homeomorphic to the 2⁢(n+1)-dimensional disc. Define

𝒞 ( n ) := { Φ ∈ C 0 ( Q r , R ( n ) , E 2 1 2 ( n ) ) ∣ Φ | ∂ ⁡ Q r , R ⁢ ( n ) = I ∂ ⁡ Q r , R ⁢ ( n ) } .

For 0<ρ<r, by using the topological degree theory, it is easily checked that Sρ+ and ∂⁡Qr,R⁢(n) link, i.e., the following hold (see [21]):

Sρ+∩∂⁡Qr,R⁢(n)=∅.
For any Φ∈𝒞(n), there holds Φ⁢(Qr,R⁢(n))∩Sρ+≠∅.

Define the min-max value

c ⁢ ( n ) := inf Φ ∈ 𝒞 ⁢ ( n ) ⁡ max ⁡ 𝔏 ⁢ ( Φ ⁢ ( Q r , R ⁢ ( n ) ) ) .

Then Lemma 4.2 and (ii) above imply

(4.5) β ρ ≤ c ⁢ ( n ) .

Observe that the inclusion I:Qr,R⁢(n)→E21/2⁢(n) belongs to 𝒞⁢(n) and that Qr,R⊂E1/2 is bounded. We deduce

(4.6) c ⁢ ( n ) ≤ max ⁡ 𝔏 ⁢ ( Q r , R ⁢ ( n ) ) ≤ sup ⁡ 𝔏 ⁢ ( Q r , R ) < ∞

because 𝔏 maps bounded sets in E1/2 into bounded sets in ℝ by (H1) and the Sobolev embedding.

Now by Remark 3.3, Lemma 4.1 and (4.5), we may use the standard deformation argument (cf. [22]) to show that c⁢(n) is a critical value of 𝔏n and thus that there exists ψn∈E21/2⁢(n) such that

d ⁢ 𝔏 n ⁢ ( ψ n ) = 0 and 𝔏 n ⁢ ( ψ n ) = c ⁢ ( n ) .

Note that (4.5) and (4.6) imply c⁢(n) to be bounded. After taking a subsequence if necessary, we may assume that c⁢(n)→c for some βρ≤c<∞. Therefore, {ψn} becomes a (PS)c*-sequence with respect to {E21/2⁢(n)}. It follows from Lemma 3.2 that 𝔏 satisfies the (PS)c∗-condition with respect to {E21/2⁢(n)}. Furthermore, taking a subsequence if necessary, we may assume that {ψn} converges to ψ in E21/2⁢(n). It is clear that ψ satisfies 𝔏⁢(ψ)=c. Since the boundedness of {ψn} in E1/2 implies that of

{ ∥ d ⁢ 𝔏 ⁢ ( ψ n ) ∥ E 1 / 2 ∗ } ,

for any φ∈E21/2⁢(n) we deduce

| 〈 d ⁢ 𝔏 ⁢ ( ψ n ) , φ - P n ⁢ φ 〉 | ≤ ∥ d ⁢ 𝔏 ⁢ ( ψ n ) ∥ E 1 / 2 ∗ ⁢ ∥ φ - P n ⁢ φ ∥ E 1 / 2 → 0

as n→∞. This and

〈 d ⁢ 𝔏 ⁢ ( ψ n ) , φ 〉 = 〈 d ⁢ 𝔏 ⁢ ( ψ n ) , P n ⁢ φ 〉 + 〈 d ⁢ 𝔏 ⁢ ( ψ n ) , φ - P n ⁢ φ 〉

= 〈 d ⁢ 𝔏 ⁢ ( ψ n ) , φ - P n ⁢ φ 〉

lead to 〈d⁢𝔏⁢(ψ),φ〉=0 for any φ∈E1/2, i.e., ψ is a critical point of 𝔏 with 𝔏⁢(ψ)≥βρ>0. The elliptic regularity theory also implies ψ∈C1⁢(M,Σ⁢M)×C1⁢(M,Σ⁢M) (cf. [2]), and thus ψ is a nontrivial classical solution of (1.1). Theorem 1.1 is proved. ∎

4.2 Proof of Theorem 1.2

We begin with the Fountain theorem for semi-definite functionals [23].

Theorem 4.3 (Fountain theorem).

Let H be a Banach space with basis {ej}j=1∞. For d≥1, we have H=H1+H2, where

H 1 = span { e j } j = 1 d 𝑎𝑛𝑑 H 2 = span { e j } j = d ∞ ¯ .

Suppose F∈C1⁢(H,R) is an even functional. For ρ>0, let

B ρ = { u ∈ H 1 ∣ ∥ u ∥ ≤ ρ } 𝑎𝑛𝑑 Γ = { γ ∈ C 0 ⁢ ( B ρ , H ) ∣ γ ⁢ ( - u ) = - γ ⁢ ( u ) ⁢ for all ⁢ u ∈ B ρ , γ | ∂ ⁡ B ρ = I ∂ ⁡ B ρ } .

Define

c := inf γ ∈ Γ ⁡ max ⁡ 𝔏 ⁢ ( γ ⁢ ( B ρ ) ) .

Suppose that F satisfies the (PS)c-condition and there exists 0<r<ρ such that

b := inf ⁡ { 𝔉 ⁢ ( u ) ∣ u ∈ H 2 , ∥ u ∥ = r } > a := max ⁡ { 𝔉 ⁢ ( u ) ∣ u ∈ H 1 , ∥ u ∥ = ρ } .

Then c is a critical value of F with c≥b.

In order to apply the Fountain theorem for the functional

𝔏 n = 𝔏 | E 2 1 / 2 ⁢ ( n ) ,

for each j≥1 we define

E 2 1 2 ( n , j ) = span ( { ψ k - } k = 1 n - l ∪ { ψ k 0 } k = 1 l ) ⊕ span { ψ k + } k = 1 j

and

E 2 1 2 + ⁢ ( n , j ) = span { ψ k + } k = j ∞ ¯ .

Then

E 2 1 2 ⁢ ( n ) = E 2 1 2 ⁢ ( n , j ) + E 2 1 2 + ⁢ ( n , j ) ,

and 𝔏n is even on E21/2⁢(n) by (H5).

Lemma 4.4.

Suppose (H2) is satisfied. Then for each j≥1 there exists ρ⁢(j)>0 such that Ln⁢(ψ)≤0, provided ψ∈E21/2⁢(n,j) and ∥ψ∥E1/2≥ρ⁢(j).

Proof.

For ψ=ψ-+ψ++ψ0∈E21/2⁢(n,j), by (3.3) and (3.14) we have

𝔏 n ⁢ ( ψ ) = ∫ 1 2 ⁢ 〈 L ⁢ ψ , ψ 〉 - H ⁢ ( x , ψ ) ⁢ d ⁢ x

≤ 1 2 ⁢ ∫ 〈 L ⁢ ψ - , ψ - 〉 ⁢ 𝑑 x + 1 2 ⁢ ∫ 〈 L ⁢ ψ + , ψ + 〉 ⁢ 𝑑 x - c 4 ⁢ ∫ | u | μ + | v | μ ⁢ d ⁢ x + c 5

(4.7) ≤ - C - ⁢ ∥ ψ - ∥ E 1 / 2 2 + λ ⁢ ( j ) 2 ⁢ ∥ ψ + ∥ 2 2 - c 4 ⁢ ∫ ( | u | μ + | v | μ ) ⁢ 𝑑 x + c 5 ,

where λ⁢(j) is the largest eigenvalue of L on E21/2⁢(n,j), and thus

1 2 ⁢ ∫ 〈 L ⁢ ψ + , ψ + 〉 ⁢ 𝑑 x ≤ λ ⁢ ( j ) 2 ⁢ ∥ ψ + ∥ 2 2 .

Since ∥u∥2≤C⁢∥u∥μ and ∥v∥2≤C⁢∥v∥μ by Hölder’s inequality, and

a λ + b λ + c λ ≤ 3 ⁢ ( a + b + c ) λ

for all nonnegative numbers a, b, c and λ≥1, we get

∥ u + ∥ 2 μ + ∥ u 0 ∥ 2 μ + ∥ u - ∥ 2 μ ≤ 3 ⁢ ( ∥ u + ∥ 2 2 + ∥ u 0 ∥ 2 2 + ∥ u - ∥ 2 2 ) μ 2

= 3 ⁢ ∥ u ∥ 2 μ ≤ C ⁢ ∥ u ∥ μ μ ,

∥ v + ∥ 2 μ + ∥ v 0 ∥ 2 μ + ∥ v - ∥ 2 μ ≤ 3 ⁢ ( ∥ v + ∥ 2 2 + ∥ v 0 ∥ 2 2 + ∥ v - ∥ 2 2 ) μ 2

= 3 ⁢ ∥ v ∥ 2 μ ≤ C ⁢ ∥ v ∥ μ μ .

From these and (4.7) we deduce

𝔏 n ⁢ ( ψ ) ≤ - C - ⁢ ∥ ψ - ∥ E 1 / 2 2 + λ ⁢ ( j ) 2 ⁢ ∥ u + ∥ 2 2 + λ ⁢ ( j ) 2 ⁢ ∥ v + ∥ 2 2 ⁢ C ⁢ ∥ u - ∥ 2 μ - C ⁢ ∥ u 0 ∥ 2 μ - C ⁢ ∥ u + ∥ 2 μ - C ⁢ ∥ v - ∥ 2 μ - C ⁢ ∥ v 0 ∥ 2 μ - C ⁢ ∥ v + ∥ 2 μ + C .

Note that dim⁢(E21/2⁢(n,j)∩E+)=j<∞. Any two norms on E21/2⁢(n,j)∩E+ are equivalent. In particular, for the L2 and the H1/2-norm on it we have constants C1, C2 (independent of n) such that

C 1 ⁢ ∥ u + ∥ 1 2 , 2 ≤ ∥ u + ∥ 2 ≤ C 2 ⁢ ∥ u + ∥ 1 2 , 2 , C 1 ⁢ ∥ v + ∥ 1 2 , 2 ≤ ∥ v + ∥ 2 ≤ C 2 ⁢ ∥ v + ∥ 1 2 , 2 .

It follows from these that

𝔏 n ⁢ ( ψ ) ≤ - C - ⁢ ∥ ψ - ∥ E 1 / 2 2 + λ ⁢ ( j ) 2 ⁢ C 2 2 ⁢ ∥ u + ∥ 1 2 , 2 2 + λ ⁢ ( j ) 2 ⁢ C 2 2 ⁢ ∥ v + ∥ 1 2 , 2 2 - C ⁢ ∥ u - ∥ 2 μ - C ⁢ ∥ u 0 ∥ 2 μ - C ⁢ C 1 μ ⁢ ∥ u + ∥ 1 2 , 2 μ

- C ⁢ ∥ v - ∥ 2 μ - C ⁢ ∥ v 0 ∥ 2 μ - C ⁢ C 1 μ ⁢ ∥ v + ∥ 1 2 , 2 μ + C .

Note that we have assumed μ>2. It is not hard to prove that there exists ρ⁢(j)>0 such that

λ ⁢ ( j ) 2 ⁢ C 2 2 ⁢ ∥ u + ∥ 1 2 , 2 2 + λ ⁢ ( j ) 2 ⁢ C 2 2 ⁢ ∥ v + ∥ 1 2 , 2 2 - C ⁢ C 1 μ ⁢ ∥ u + ∥ 1 2 , 2 μ - C ⁢ C 1 μ ⁢ ∥ v + ∥ 1 2 , 2 μ ≤ 0

for all ψ∈E21/2⁢(n,j) with ∥ψ∥E1/2≥ρ⁢(j). Hence, 𝔏n⁢(ψ)≤0. ∎

Note that E21/2+⁢(n,j) does not depend on n. For 1≤p<2∗, let us define

β j , p := sup ⁡ { ∥ ψ ∥ p ∣ ψ ∈ E 2 1 2 + ⁢ ( n , j ) , ∥ ψ ∥ E 1 / 2 = 1 } .

Lemma 4.5.

β j , p → 0 as j→∞.

Proof.

By the definition of βj,p, for each j we can find ψj∈E21/2+⁢(n,j) such that ∥ψj∥E1/2=1 and 12⁢βj,p<∥ψj∥p. Using the compactness of the embedding E1/2↪Lp×Lp, we may assume (after taking a subsequence if necessary) ψj⇀ψ in E1/2 and ψj→ψ in Lp×Lp for some ψ∈E1/2. We can also assume ψj→ψ in L2×L2 since the embedding E1/2↪L2×L2 is compact. For a fixed

ψ ^ ∈ { ψ k - } k = 1 ∞ ∪ { ψ k 0 } k = 1 l ∪ { ψ k + } k = 1 ∞ ,

from

ψ j ∈ E 2 1 / 2 + ⁢ ( n , j ) = span { ψ k + } k = j ∞ ¯

we derive that (ψj,ψ^)2=0 for sufficiently large j, and so limj→∞⁡(ψj,ψ^)2=0. These show that ψj⇀0 in L2×L2. Hence ψ=0, and therefore

1 2 ⁢ β j , p < ∥ ψ j ∥ p → 0 . ∎

Lemma 4.6.

Under conditions (H1) and (H2), there exists r⁢(j)>0 such that

b ⁢ ( j ) := inf ⁡ { 𝔏 n ⁢ ( ψ ) ∣ ψ ∈ E 2 1 2 + ⁢ ( n , j ) , ∥ ψ ∥ E 1 / 2 = r ⁢ ( j ) } → ∞

as j→∞.

Proof.

By (2.9) and (3.6), we have for ψ∈E21/2+⁢(n,j) with ∥ψ∥E1/2=r⁢(j),

𝔏 n ⁢ ( ψ ) = 1 2 ⁢ ∫ M 〈 ψ , L ⁢ ψ 〉 ⁢ 𝑑 x - ∫ M H ⁢ ( x , ψ ) ⁢ 𝑑 x

≥ C + ⁢ ∥ ψ ∥ E 1 2 2 - c 2 ⁢ ∫ M ( | u | p + | v | q ) ⁢ 𝑑 x - c 3

≥ C + ⁢ ∥ ψ ∥ E 1 2 2 - c 2 ⁢ ∥ ψ ∥ p p - c 2 ⁢ ∥ ψ ∥ q q - c 3

≥ C + ⁢ r ⁢ ( j ) 2 - c 2 ⁢ r ⁢ ( j ) p ⁢ β j , p p - c 2 ⁢ r ⁢ ( j ) q ⁢ β j , q q - c 3 .

Without loss of generality, we may assume p>q, and proceed in two cases.

Case 1. If 0<r⁢(j)<1, then

𝔏 n ⁢ ( ψ ) ≥ C + ⁢ r ⁢ ( j ) 2 - c 2 ⁢ r ⁢ ( j ) q ⁢ β j , p p - c 2 ⁢ r ⁢ ( j ) q ⁢ β j , q q - c 3 .

Since the function

r ⁢ ( j ) → C + ⁢ r ⁢ ( j ) 2 - c 2 ⁢ r ⁢ ( j ) q ⁢ β j , p p - c 2 ⁢ r ⁢ ( j ) q ⁢ β j , q q - c 3

attains the maximum

( C + - 2 ⁢ C + q ) ⁢ ( 2 ⁢ C + c 2 ⁢ q ⁢ ( β j , p p + β j , q q ) ) 2 q - 2 - c 3

r ⁢ ( j ) = ( 2 ⁢ C + q ⁢ c 2 ⁢ ( β j , p p + β j , q q ) ) 1 q - 2 ,

we may use Lemma 4.5 to obtain

b ⁢ ( j ) ≥ ( C + - 2 ⁢ C + q ) ⁢ ( 2 ⁢ C + c 2 ⁢ q ⁢ ( β j , p p + β j , q q ) ) 2 q - 2 - c 3 → ∞

as j→∞.

Case 2. If r⁢(j)>1, we have

𝔏 n ⁢ ( ψ ) ≥ C + ⁢ r ⁢ ( j ) 2 - c 2 ⁢ r ⁢ ( j ) p ⁢ β j , p p - c 2 ⁢ r ⁢ ( j ) p ⁢ β j , q q - c 3 .

As in Case 1, we may use Lemma 4.5 to prove

b ⁢ ( j ) ≥ ( C + - 2 ⁢ C + p ) ⁢ ( 2 ⁢ C + c 2 ⁢ p ⁢ ( β j , p p + β j , q q ) ) 2 p - 2 - c 3 → ∞

as j→∞. The lemma is proved. ∎

Proof of Theorem 1.2.

Note that ρ⁢(j) in Lemma 4.4 can be replaced by a larger number. By taking ρ⁢(j) large if necessary, we may assume that r⁢(j)<ρ⁢(j) for each j. Then from Remark 3.3, Lemma 4.4, Lemma 4.6 and Theorem 4.3 we may derive that for large j (independent of n) the min-max value

c ⁢ ( n , j ) := inf γ ∈ Γ ⁢ ( n , j ) ⁡ max ⁡ 𝔏 n ⁢ ( γ ⁢ ( B ⁢ ( n , j ) ) )

is a critical value of 𝔏n, where

B ⁢ ( n , j ) = { ψ ∈ E 2 1 2 ⁢ ( n , j ) ∣ ∥ ψ ∥ E 1 / 2 = ρ ⁢ ( j ) }

and

Γ ⁢ ( n , j ) = { γ ∈ C 0 ⁢ ( B ⁢ ( n , j ) , E 2 1 2 ⁢ ( n ) ) ∣ γ ⁢ ( - ψ ) = - γ ⁢ ( ψ ) , γ | ∂ ⁡ B ⁢ ( n , j ) = I | ∂ ⁡ B ⁢ ( n , j ) } .

Let

B ( j ) = { ψ ∈ E 1 1 2 ⊕ span { ψ k + } k = 1 j ∣ ∥ ψ ∥ E 1 / 2 ≤ ρ ( j ) } .

Observe that 𝔏 is bounded on each bounded subset in E1/2. We obtain

b ( j ) ≤ c ( n , j ) ≤ max 𝔏 n ( B ( n , j ) ) ≤ sup 𝔏 ( B ( j ) ) = : d ( j ) < ∞ .

Furthermore, passing to a subsequence if necessary, we may assume

c ⁢ ( n , j ) → c ⁢ ( j ) ∈ [ b ⁢ ( j ) , d ⁢ ( j ) ] as ⁢ n → ∞ .

Note that c⁢(n,j) is a critical value of 𝔏n. We have ψn,j∈E21/2⁢(n) such that d⁢𝔏n⁢(ψn,j)=0 and 𝔏n(ψn,j))=c(n,j). Moreover, since 𝔏 satisfies the (PS)c⁢(j)∗-condition with respect to {E21/2⁢(n)}, after taking a subsequence if necessary, there exists ψj∈E1/2 such that ψn,j→ψj in E1/2 as n→∞. As in the proof of Theorem 1.1, we can prove that ψj is a critical point of 𝔏 with 𝔏⁢(ψj)=c⁢(j). Finally, by noting that b⁢(j)≤c⁢(j) and that b⁢(j)→∞ by Lemma 4.6, the proof is complete. ∎

Communicated by Yiming Long

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 10971014

Award Identifier / Grant number: 11271044

Funding statement: This work was partially supported by the National Natural Science Foundation of China through grants 10971014 and 11271044.

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Received: 2017-03-11

Revised: 2017-07-23

Accepted: 2017-09-18

Published Online: 2017-10-18

Published in Print: 2018-02-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/ans-2017-6034

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Dirac System; Compact Spin Manifolds; Variational Methods

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