Multiplicity and concentration of semi-classical solutions to nonlinear Dirac-Klein-Gordon systems

Yanheng Ding; Yuanyang Yu; Xiaojing Dong

doi:10.1515/ans-2022-0011

Open Access Published by De Gruyter July 2, 2022

Multiplicity and concentration of semi-classical solutions to nonlinear Dirac-Klein-Gordon systems

Yanheng Ding , Yuanyang Yu and Xiaojing Dong

From the journal Advanced Nonlinear Studies

https://doi.org/10.1515/ans-2022-0011

Abstract

In the present article, we study multiplicity of semi-classical solutions of a Yukawa-coupled massive Dirac-Klein-Gordon system with the general nonlinear self-coupling, which is either subcritical or critical growth. The number of solutions obtained is described by the ratio of maximum and behavior at infinity of the potentials. We use the variational method that relies upon a delicate cutting off technique. It allows us to overcome the lack of convexity of the nonlinearities.

Keywords: Dirac-Klein-Gordon system; semi-classical states; multiplicity; subcritical and critical nonlinearities

MSC 2010: 35Q40; 49J35

1 Introduction and main results

In this article, we study the solitary wave solutions of the massive Dirac-Klein-Gordon system involving an external self-coupling:

(1.1) i ℏ ∂ t ψ + i c ℏ ∑ k = 1 3 α k ∂ k ψ − m c 2 β ψ − λ ( x ) ϕ β ψ = g ( x , ∣ ψ ∣ ) ψ , ℏ 2 c 2 ∂ t 2 ϕ − ℏ 2 Δ ϕ + M c 2 ϕ = 4 π λ ( x ) ( β ψ ) ⋅ ψ

for ( t , x ) ∈ R × R 3 , where c is the speed of light, ℏ is Planck’s constant, λ ( x ) > 0 is the coupling potential, m is the mass of the electron, and M is the mass of the meson (we use the notation u ⋅ v to express the inner product of u , v ∈ C 4 ). Here α 1 , α 2 , α 3 , and β are 4 × 4 complex Pauli matrices:

β = I 0 0 − I , α k = 0 σ k σ k 0 , k = 1 , 2 , 3 ,

with

σ 1 = 0 1 1 0 , σ 2 = 0 − i i 0 , σ 3 = 1 0 0 − 1 .

System (1.1) has been derived as mathematical models of particle physics, especially in nonlinear topics. This system is inspired by approximate descriptions of the external force involving only functions of fields, which describes the Dirac and Klein-Gordon equations coupled through the Yukawa interaction between a Dirac field ψ ∈ C 4 and a scalar field ϕ ∈ R (see [5]). The nonlinear self-coupling g ( x , ∣ ψ ∣ ) ψ describes a self-interaction in Quantum electrodynamics and gives a closer description of many particles found in the real world, which has been widely considered in the literature (see [23,24,26] etc. and references therein).

There has been many works on the existence of solutions of system (1.1). When there is no external self-coupling, i.e., g ≡ 0 , (1.1) has been studied for a long time and results are available concerning the cauchy problem (see [6,9,10, 28,32]). In [9], Chadam obtained the first result on the global existence and uniqueness of solutions of (1.1) (in one space dimension) under suitable assumptions on the initial data. For later developments, Chadam and Glassey [10] yielded the existence of a global solution in three space dimensions. By using classical Strichartz-type time-space estimates, low regularity solutions of the Dirac-Klein-Gordon system were obtained by Bournaveas in [6]. With respect to the nonautonomous system (1.1) with external self-coupling, Ding and Xu [13] were devoted to the existence and concentration phenomenon of stationary semi-classical solutions for the subcritical nonlinearities. In addition, for the critical nonlinearities, Ding et al. [16] obtained the same results. Here, stationary solution means a solution of the type

(1.2) ψ ( t , x ) = φ ( x ) e − i ξ t / ℏ , ξ ∈ R , φ : R 3 → C 4 ϕ = ϕ ( x ) .

As far as the multiplicity of solutions of system (1.1) is concerned, there is a pioneering work by Esteban et al. (see [21]) in which a multiplicity result is studied by using the variational arguments. We emphasize that this work was mainly concerned with the autonomous system. Besides, limited work has been done in the semi-classical approximation. For small ℏ , the solitary waves are referred to as semi-classical states.

Motivated by the works just described above, we are interested in the multiplicity of stationary semi-classical solutions to system (1.1) with some general subcritical self-coupling nonlinearity f ( x , ∣ φ ∣ ) φ = W ( x ) ∣ φ ∣ p − 2 φ , p ∈ ( 2 , 3 ) or critical self-coupling nonlinearity f ( x , ∣ φ ∣ ) = W 1 ( x ) ∣ φ ∣ p − 2 φ + W 2 ( x ) ∣ φ ∣ φ . We shall see that one of the difficulties with nonlinear Dirac-Klein-Gordon system (1.1) is caused by the spectra of Dirac operator which are unbounded and consist of essential spectra. A further problem is caused by the Klein-Gordon equations. In [14], Ding and Ruf overcame the first difficulty with nonlinear Maxwell-Dirac system by variational methods for strongly indefinite problems. However, because the Klein-Gordon equations are also influenced by Planck’s constant ℏ , the limit equation is different between the Dirac-Klein-Gordon systems and the Maxwell-Dirac systems. As a result, the method of nonlinear Maxwell-Dirac systems cannot be directly applied to the nonlinear Dirac-Klein-Gordon systems (1.1). Last but not the least, (1.1) involves several different potentials which bring a competition between the potentials W ( x ) and λ ( x ) or between the potentials W 1 ( x ) , W 2 ( x ) and λ ( x ) . This will affect our important result – the number of solutions to Dirac-Klein-Gordon systems.

For notational convenience, denoted by c = 1 , ε = ℏ , α = ( α 1 , α 2 , α 3 ) , and α ⋅ ∇ = ∑ k = 1 3 α k ∂ k , we are concerned with (substitute (1.1) in (1.2)) the following stationary nonlinear Dirac-Klein-Gordon system:

(1.3) i c ε α ⋅ ∇ φ − a β φ + ω φ − λ ( x ) ϕ β φ = f ( x , ∣ φ ∣ ) φ , − ε 2 Δ ϕ + M c 2 ϕ = 4 π λ ( x ) ( β φ ) ⋅ φ ,

where a = m c 2 > 0 and ω ∈ R .

Case (A): The subcritical case

First we deal with the subcritical case:

f ( x , ∣ φ ∣ ) φ = W ( x ) ∣ φ ∣ p − 2 φ , p ∈ ( 2 , 3 ) .

Setting

κ ≔ max x ∈ R 3 W ( x ) , κ ∞ ≔ limsup ∣ x ∣ → ∞ W ( x ) , W ≔ { x ∈ R 3 : W ( x ) = κ } ,

we assume the nonlinear potential satisfies:

( W ) W ∈ C 0 , 1 ( R 3 ) with inf W > 0 and κ ∞ < κ .

In addition, setting

λ ¯ ≔ max x ∈ R 3 λ ( x ) , λ ∞ ≔ limsup ∣ x ∣ → ∞ λ ( x ) , Γ ≔ { x ∈ R 3 : λ ( x ) = λ ¯ } ,

we make the following hypotheses:

( λ ) λ ( x ) ∈ C 0 , 1 ( R 3 ) with inf λ ( x ) > 0 and λ ∞ < λ ¯ .

Theorem 1.1

Assume that ω ∈ ( − a , a ) , W ∩ Γ ≠ ∅ , and ( W ) , ( λ ) are satisfied. Then there is a constant λ 0 > 0 , such that for any λ ¯ ≤ λ 0 and m ∈ N with

m ≤ m ( ∞ ) ≔ min λ ¯ λ ∞ , κ κ ∞ 2 p − 2 ,

there is ε m > 0 such that (1.3) has at least m pairs of semi-classical solutions ( φ ε , ϕ ε ) ∈ ⋂ q ≥ 2 W 1 , q ( R 3 , C 4 ) × C 2 ( R 3 , R ) provided ε ≤ ε m .

Remark 1.1

If additionally ∇ λ ( x ) and ∇ W ( x ) are bounded, then among the solutions, the ground state (the least energy solution) denoted by w ε satisfies (see [13]):

There is a maximum point x ε of ∣ φ ε ∣ with lim ε → 0 dist ( x ε , W ∩ Γ ) = 0 such that the pair ( u ε , V ε ) , where u ε ( x ) ≔ φ ε ( ε x + x ε ) and V ε ≔ ϕ ε ( ε x + x ε ) , converges in H 1 × H 1 to a ground state solution of (the limit equation)

(1.4) i c α ⋅ ∇ u − a β u + ω u − λ ¯ V β u = κ ∣ u ∣ p − 2 u − Δ V + M c 2 V = 4 π λ ¯ ( β u ) ⋅ u ;
There exist C 1 , C 2 such that ∣ φ ε ( x ) ∣ ≤ C 1 exp − C 2 ε ∣ x − x ε ∣ .

Moreover, for large c , it is called the nonrelativistic limit problem. With Theorem 1.1 and [18,20], we obtain the following corollary.

Corollary 1.1

Let ν > 0 , p ∈ 12 5 , 8 3 . Assume 0 < − ω < m c 2 , m c 2 + ω → ν m , as c → + ∞ , ω → − ∞ . Under the assumption of Theorem 1.1 and Remark 1.1, the ground solution ( φ c , ϕ c ) where φ c = ( u c , v c ) of equation (1.4) have the following properties:

u c and ϕ c converge in H 1 to 0;
v c converges in H 1 to a solution of a coupled system of nonlinear Schrödinger-type equations
− Δ v 1 + 2 ν v 1 = 2 m κ ∣ v ∣ p − 2 v 1 , − Δ v 2 + 2 ν v 2 = 2 m κ ∣ v ∣ p − 2 v 2 ,

as c → + ∞ , ω → − ∞ , where v = ( v 1 , v 2 ) : R 3 → C 2 is the wave function corresponding to positive energy values, and ν > 0 is a constant.

Remark 1.2

Corollary 1.1 states that the semi-classical limit equation of Dirac-Klein-Gordon systems is the autonomuous Dirac-Klein-Gordon systems and the nonrelativistic limit equation of the autonomuous Dirac-Klein-Gordon systems is a coupled system of nonlinear Schrödinger-type equations. However, we do not know whether the semi-classical limit equation of the nonrelativistic limit equation of Dirac-Klein-Gordon systems is a coupled system of nonlinear Schrödinger-type equations or not.

Case (B): The critical case

Next we deal with the critical case:

f ( x , ∣ φ ∣ ) φ = W 1 ( x ) ∣ φ ∣ p − 2 φ + W 2 ( x ) ∣ φ ∣ φ , p ∈ ( 2 , 3 ) .

Denoting, for j = 1 , 2 ,

κ j ≔ max x ∈ R 3 W j ( x ) , κ j ∞ ≔ limsup ∣ x ∣ → ∞ W j ( x ) , W j ≔ { x ∈ R 3 : W j ( x ) = κ j } ,

we assume the nonlinear potentials satisfy:

( W → 1 ) W j ∈ C 0 , 1 ( R 3 ) with inf W j > 0 , κ j ∞ < κ j .

Let S be the best Sobolev constant: S ∣ u ∣ 6 2 ≤ ∣ ∇ u ∣ 2 2 for all u ∈ H 1 ( R 3 , R ) and τ p be the least energy of the following subcritical equation:

i α ⋅ ∇ u − a β u − ω u = ∣ u ∣ p − 2 u

(which always exists, see [11]). Assume furthermore that

( W → 2 ) There holds

κ 1 ∞ − 1 κ 2 ∞ p − 2 < S 3 2 6 τ p p − 2 2 .

Setting

m ( ∞ → ) ≔ min λ ¯ λ ∞ , κ 1 κ 1 ∞ 2 p − 2 , κ 2 κ 2 ∞ 2 .

Theorem 1.2

Assume that ω ∈ ( − a , a ) , W 1 ∩ W 2 ∩ Γ ≠ ∅ , and ( λ ) , ( W → 1 ) , ( W → 2 ) are satisfied. Then there is a constant λ 0 > 0 , such that for any λ ¯ ≤ λ 0 , m ∈ N with m ≤ m ( ∞ → ) there is ε m > 0 such that (1.3) has at least m pairs of semi-classical solutions ( φ ε , ϕ ε ) ∈ ⋂ q ≥ 2 W 1 , q ( R 3 , C 4 ) × C 2 ( R 3 , R ) provided ε ≤ ε m .

Remark 1.3

If additionally ∇ λ and ∇ W j ( j = 1 , 2 ) are bounded, then among the solutions, the ground state (the least energy solution) denoted by w ε satisfies (see [16]):

There is a maximum point x ε of ∣ φ ε ∣ with lim ε → 0 dist ( x ε , W 1 ∩ W 2 ∩ Γ ) = 0 such that the pair ( u ε , V ε ) , where u ε ( x ) ≔ φ ε ( ε x + x ε ) and V ε ≔ ϕ ε ( ε x + x ε ) , converges in H 1 × H 1 to a ground state solution of (the limit equation)

(1.5) i α ⋅ ∇ u − a β u + ω u − λ ¯ V β u = κ 1 ∣ u ∣ p − 2 u + κ 2 ∣ u ∣ u − Δ V + M V = 4 π λ ¯ ( β u ) ⋅ u ;
There exist C 1 , C 2 such that ∣ φ ε ( x ) ∣ ≤ C 1 exp − C 2 ε ∣ x − x ε ∣ .

It is standard that (1.3) is equivalent to, letting ( u ( x ) , V ( x ) ) = ( φ ( ε x ) , ϕ ( ε x ) ) ,

i α ⋅ ∇ u − a β u + ω u − λ ε ( x ) V β u = f ( ε x , ∣ u ∣ ) u , − Δ V + M V = 4 π λ ε ( x ) ( β u ) ⋅ u ,

where λ ε ( x ) = λ ( ε x ) , W ε ( x ) = W ( ε x ) . We will in the sequel focus on this equivalent problem. Our proofs are variational: the semi-classical solutions that are associated with the equivalent problem (1) are obtained as critical points of an energy functional Φ ε .

Remark 1.4

Every pair of semi-classical solutions above that has different energy by using variational methods for strongly indefinite problems converge to a ground state solution of the limit equation (1.4) or (1.5), respectively.

There have been many works on multiplicity of semi-classical states of nonlinear Schrödinger-Poisson systems arising in nonrelativistic quantum mechanics, see for example [1,2,3] and their references. Later, Ding and Ruf [14] studied nonlinear Maxwell-Dirac systems, which obtained the multiplicity of semi-classical states under the assumptions of subcritical or critical nonlinearities, respectively. Recently, Ding et al. yielded the multiplicity of semi-classical states for nonlinear Dirac equations of space-dimension n in [17]. It is worth noting that the limit equation of the nonlinear Maxwell-Dirac systems or Dirac equations is a single equation. The problems in Dirac-Klein-Gordon systems are difficult because the limit equation is a system rather than a single equation, which is going to limit the number of solutions. As far as the authors know, there have been few results on the multiplicity of semiclassical solutions to nonlinear Dirac-Klein-Gordon systems.

An outline of this article is as follows: In Section 2, we treat the linking argument which gives us a min-max scheme. In Section 3, we study the limit equation and introduce the cut-off arguments. Finally, in Section 4, the combination of the results in Sections 2 and 3 proves Theorem 1.1, Corollary 1.1, and Theorem 1.2.

2 The variational framework

In the absence of nonrelativistic limit, the speed of light c is a constant. Without loss of generality, we assume c = 1 . Our arguments for the problem are variational, so we start from discussing the proper variational setting. Throughout the article we always let the hypotheses of Theorem 1.1 be satisfied. Since W ∩ Γ ≠ ∅ , without loss of generality, we assume that 0 ∈ W ∩ Γ in the subcritical case and 0 ∈ W 1 ∩ W 2 ∩ Γ in the critical case.

In the sequel, we denote by ∣ ⋅ ∣ q the usual L q -norm and by ( ⋅ , ⋅ ) q the usual L q -inner product. Let H ω = i α ⋅ ∇ − a β + ω denote the self-adjoint operator on L 2 ≡ L 2 ( R 3 , C 4 ) with domain D ( H ω ) = H 1 ≡ H 1 ( R 3 , C 4 ) . It is well known that σ ( H ω ) = σ c ( H ω ) = R \ ( − a + ω , a + ω ) , where σ ( ⋅ ) and σ c ( ⋅ ) denote the spectrum and the continuous spectrum. For ω ∈ ( − a , a ) , the space L 2 possesses the orthogonal decomposition:

(2.1) L 2 = L + ⊕ L − , u = u + + u − ,

so that H ω is positive definite (resp. negative definite) in L + (resp. L − ). Let E ≔ D ( ∣ H ω ∣ 1 2 ) = H 1 2 be equipped with the inner product

( u , v ) = R ( ∣ H ω ∣ 1 2 u , ∣ H ω ∣ 1 2 v ) 2

and the induced norm ‖ u ‖ = ( u , v ) 1 2 , where ∣ H ω ∣ and ∣ H ω ∣ 1 2 , denote, respectively, the absolute value of H ω and the square root of ∣ H ω ∣ . Since σ ( H ω ) = R \ ( − a + ω , a + ω ) , one has

(2.2) ( a − ∣ ω ∣ ) ∣ u ∣ 2 2 ≤ ‖ u ‖ 2 for all u ∈ E .

Note that this norm is equivalent to the usual H 1 2 -norm, hence E embeds continuously into L q for all q ∈ [ 2 , 3 ] and compactly into L loc q for all q ∈ [ 1 , 3 ) . It is clear that E possesses the following orthogonal decomposition:

(2.3) E = E + ⊕ E − where E ± = E ∩ L ± ,

with respect to both ( ⋅ , ⋅ ) 2 and ( ⋅ , ⋅ ) inner products. This decomposition induces also a natural decomposition of L p , hence there is d p > 0 such that

(2.4) d p ∣ u ± ∣ p p ≤ ∣ u ∣ p p for all u ∈ E .

Let H 1 ( R 3 , R ) be equipped with the equivalent norm

‖ v ‖ H 1 = ∫ R 3 ∣ ∇ v ∣ 2 + M v 2 d x 1 / 2 , ∀ v ∈ H 1 ( R 3 , R ) .

Then (1.5) can be reduced to a single equation with a nonlocal term. Actually, for any v ∈ H 1 ( R 3 , R ) ,

(2.5) 4 π ∫ R 3 λ ε ( x ) ( β u ) u ⋅ v d x ⩽ 4 π λ ¯ ∫ R 3 ∣ u ∣ 2 ∣ v ∣ d x ⩽ 4 π λ ¯ ∣ u ∣ 12 / 5 2 ∣ v ∣ 6 ⩽ 4 π λ ¯ S − 1 / 2 ∣ u ∣ 12 / 5 2 ‖ v ‖ H 1 .

Hence, there exists a unique V ε , u ∈ H 1 ( R 3 , R ) such that

∫ R 3 ∇ V ε , u ⋅ ∇ z + M ⋅ V ε , u z d x = 4 π ∫ R 3 λ ε ( x ) ( β u ) u ⋅ z d x

for all z ∈ H 1 ( R 3 , R ) . It follows that V ε , u satisfies the Schrödinger-type equation

− Δ V ε , u + M ⋅ V ε , u = 4 π λ ε ( x ) ( β u ) u

and there holds

(2.6) V ε , u ( x ) = ∫ R 3 λ ε ( y ) [ ( β u ) u ] ( y ) ∣ x − y ∣ e − M ∣ x − y ∣ d y .

Substituting V ε , u in (1.5), we are led to the equation

(2.7) H ω u − λ ε ( x ) V ε , u β u = f ( ε x , u ) .

On E we define the functional

Φ ε ( u ) = 1 2 ( ∥ u + ∥ 2 − ∥ u − ∥ 2 ) − Γ λ ε ( u ) − Ψ ε ( u )

for u = u + + u − , where

Γ λ ε ( u ) = 1 4 ∫ R 3 λ ε ( x ) V ε , u ⋅ ( β u ) u d x = 1 4 ∬ λ ε ( x ) [ ( β u ) u ] ( x ) λ ε ( y ) [ ( β u ) u ] ( y ) ∣ x − y ∣ e − M ∣ x − y ∣ d y d x

and

Ψ ε ( u ) = ∫ R 3 F ( ε x , ∣ u ∣ ) d x where F ( x , ∣ u ∣ ) = ∫ 0 ∣ u ∣ f ( x , s ) s d s .

It follows by standard arguments that Φ ε ∈ C 2 ( E , R ) and any critical point of Φ ε is a solution of (1).

Before going on we observe the following [13]:

Lemma 2.1

One has: for any u , v ∈ E

For every ε > 0 , Γ λ ε is nonnegative and weakly sequentially lower semi-continuous.
Γ λ ε ′ ( u ) v = 1 2 ℜ ∬ λ ε ( x ) λ ε ( y ) e − M ∣ x − y ∣ ∣ x − y ∣ ( [ ( β u ) u ] ( x ) [ ( β u ) v ] ( y ) + [ ( β u ) u ] ( y ) [ ( β u ) v ] ( x ) ) d y d x = ∫ R 3 λ ε ( x ) V ε , u ⋅ ℜ ( β u ) v d x ;
If u n ⇀ u in E , then Γ λ ε ( u n ) − Γ λ ε ( u n − u ) → Γ λ ε ( u ) and Γ λ ε ′ ( u n ) − Γ λ ε ′ ( u n − u ) → Γ λ ε ′ ( u ) ;
∣ Γ λ ε ( u ) ∣ ≤ S − 1 λ ¯ 2 ∣ u ∣ 12 5 2 ≤ C 1 λ ¯ 2 ‖ u ‖ 4 ; ∣ Γ λ ε ′ ( u ) v ∣ ≤ 4 π S − 1 λ ¯ 2 ∣ u ∣ 3 2 ∣ u ∣ 2 ∣ v ∣ 2 ≤ C 2 λ ¯ 2 ‖ u ‖ 3 ‖ v ‖ ; ∣ Γ λ ε ″ ( u ) [ v , v ] ∣ ⩽ C 3 λ ¯ 2 ‖ u ‖ 2 ‖ v ‖ 2 ,
where here (and below) by C j we denote a generic positive constant.

We will study the multiplicity of critical points of Φ ε via linking arguments. Setting E m = E − ⊕ H m for any finite dimensional subspace H m ⊂ E + with dim H m = m .

Lemma 2.2

The following conclusions are true:

There are r > 0 and τ > 0 , both independent of ε , such that Φ ε ∣ B r + ≥ 0 and Φ ε ∣ S r + ≥ τ where B r + = B r ∩ E + = { u ∈ E + : ‖ u ‖ ≤ r } and S r + = ∂ B r + = { u ∈ E + : ‖ u ‖ = r } ;
For any finite dimensional subspace H m ⊂ E + , there exist C = C m > 0 and R = R m > r both independent of ε , such that, for all ε > 0 , max Φ ε ( E m ) ≤ C , and Φ ε ( u ) < 0 for all u ∈ E m \ B R .

Proof

The subcritical case can be checked easily (see [13]). We verify the critical case. Recall that ∣ u ∣ p p ≤ d ¯ p ‖ u ‖ p for all u ∈ E by Sobolev embedding theorem. 1 ∘ follows easily because, for u ∈ E + and δ > 0 small enough

Φ ε ( u ) = 1 2 ‖ u ‖ 2 − Γ λ ε ( u ) − Ψ ε ( u ) ≥ 1 2 ‖ u ‖ 2 − c 1 λ ¯ 2 ‖ u ‖ 4 − ∣ W 1 ∣ ∞ p ∣ u ∣ p p − ∣ W 2 ∣ ∞ 3 ∣ u ∣ 3 3 ≥ 1 2 ‖ u ‖ 2 − c 1 λ ¯ 2 ‖ u ‖ 4 − d ¯ p ∣ W 1 ∣ ∞ p ‖ u ‖ p − d ¯ 3 ∣ W 2 ∣ ∞ 3 ‖ u ‖ 3 ,

with c 1 independent of u and p > 2 .

For checking 2 ∘ , take a finite dimensional subspace H m ⊂ E + . In virtue of (2.4), for u = u − + u + ∈ E m = E − + H m where u + ∈ H m , u − ∈ E − , one obtains

Φ ε ( u ) = 1 2 ‖ u + ‖ 2 − 1 2 ‖ u − ‖ 2 − Γ λ ε ( u ) − Ψ ε ( u ) ≤ 1 2 ‖ u + ‖ 2 − 1 2 ‖ u − ‖ 2 − inf W 2 3 ∣ u + ∣ 3 3 ≤ 1 2 ‖ u + ‖ 2 − 1 2 ‖ u − ‖ 2 − d ¯ 3 inf W 2 3 c ‖ u + ‖ 3 ,

proving the conclusions.□

In particular, for any e ∈ E + \ { 0 } , setting H 1 = R e and E e = E − ⊕ H 1 , the conclusion 2 ∘ holds. Based on this lemma, for any ε ≥ 0 , let c ε denote the minimax level of Φ ε deduced by the linking structure [30]:

(2.8) c ε ≔ inf e ∈ E + \ { 0 } max u ∈ E e Φ ε ( u ) = inf e ∈ E + \ { 0 } max u ∈ E ˆ e Φ ε ( u ) ,

where E ˆ e = E − ⊕ R + e .

Recall that a sequence { u n } ⊂ E is called to be a ( C ) c -sequence for Φ ∈ C 1 ( E , R ) if Φ ( u n ) → c and ( 1 + ∥ u n ∥ ) Φ ′ ( u n ) → 0 . We say that Φ satisfies the ( C ) c -condition if any ( C ) c -sequence for Φ has a convergent subsequence. Below we are going to study ( C ) c -sequences for Φ ε .

Lemma 2.3

For every pair of constants c 1 , c 2 > 0 , there exists a constant Λ > 0 , depending only on c 1 , c 2 , λ ¯ , such that for any u ∈ E with

(2.9) ∣ Φ ε ( u ) ∣ ≤ c 1 and ‖ u ‖ ⋅ ‖ Φ ε ′ ( u ) ‖ ≤ c 2 ,

we have

∥ u ∥ ≤ Λ .

Furthermore, Λ is an increasing function with respect to λ ¯ > 0 .

Proof

Again we only check the critical case because the subcritical case is easier than the critical case and is similar to [13].

Take u ∈ E such that (2.9) is satisfied. Without loss of generality we may assume that ‖ u ‖ ≥ 1 . The form of Φ ε and Lemma 2.1 imply that

c 1 + c 2 ≥ Φ ε ( u ) − 1 2 Φ ε ′ ( u ) u = Γ λ ε ( u ) − 1 p − 1 2 ∫ R 3 W 1 ( ε x ) ∣ u ∣ p d x + 1 6 ∫ R 3 W 2 ( ε x ) ∣ u ∣ 3 d x ≥ 1 2 − 1 p inf W 1 ∣ u ∣ p p + inf W 2 6 ∣ u ∣ 3 3 ,

that is,

(2.10) ∣ u ∣ 3 ≤ C 1 and ∣ u ∣ p ≤ C 2 .

By [13, Lemma 2.4], we have V u λ ¯ ‖ u ‖ 6 ≤ ∣ u ∣ p , p ∈ ( 2 , 3 ) , which, together with the Hölder inequality, implies that

(2.11) R ∫ λ ε ( x ) V ε , u ( β u ) ⋅ ( u + − u − ) ≤ λ ¯ ‖ u ‖ ℜ ∫ V ε , u ‖ u ‖ ( β u ) ⋅ ( u + − u − ) ≤ λ ¯ 2 ‖ u ‖ V ε , u λ ¯ ‖ u ‖ 6 ∣ u ∣ p ⋅ ∣ u + − u − ∣ q ≤ λ ¯ 2 C 3 ‖ u ‖ ⋅ ∣ u ∣ q .

Let q = 6 p 5 p − 6 . Then 2 < q < 3 and 1 p + 1 q + 1 6 = 1 . Set

ξ = 0 , if q = p , 2 ( p − q ) q ( p − 2 ) if q < p , 3 ( q − p ) q ( 3 − p ) if q > p ,

we deduce that ξ < 1 and

∣ u ∣ q ≤ ∣ u ∣ 2 ξ ⋅ ∣ u ∣ p 1 − ξ , if 2 < q ≤ p , ∣ u ∣ 3 ξ ⋅ ∣ u ∣ p 1 − ξ if p < q < 3 .

This, together with Lemma 2.1 and (2.11), implies that

(2.12) Γ λ ε ′ ( u ) ( u + − u − ) ∣ ≤ λ ¯ 2 C 4 ‖ u ‖ 1 + ξ .

Then (2.10) and (2.12) imply that

c 2 ≥ Φ ε ′ ( u ) ( u + − u − ) = ‖ u ‖ 2 − Γ λ ε ′ ( u ) ( u + − u − ) − R ∫ R 3 W 1 ( ε x ) ∣ u ∣ p − 2 u ( u + − u − ) d x − R ∫ R 3 W 2 ( ε x ) ∣ u ∣ u ( u + − u − ) d x ≥ ‖ u ‖ 2 − λ ¯ 2 C 4 ‖ u ‖ 1 + ξ − κ 1 C 2 p − κ 2 C 1 3 .

That is,

(2.13) ‖ u ‖ 2 ≤ λ ¯ 2 C 4 ‖ u ‖ 1 + ξ + C 5 .

Therefore, there is Λ = Λ ( c 1 , c 2 , λ ) such that

‖ u ‖ ≤ Λ .

Moreover, (2.13) implies Λ is increasing in λ ¯ .□

Lemma 2.3 has an immediate consequence, which implies the boundedness of a ( C ) c -sequence:

Corollary 2.1

Consider ε ∈ ( 0 , 1 ] , and { u n ε } is the corresponding ( C ) c -sequence for Φ ε . If there exists C > 0 such that ∣ c ε ∣ ⩽ C for all ε , then we have (up to a subsequence if necessary)

∥ u n ε ∥ ≤ Λ ,

where Λ found in Lemma 2.3 depends on λ and the pair c 1 = C and c 2 = 1 .

Additionally, for later aims we define the operator V : E → H 1 ( R 3 , R ) by V ( u ) = V u . According to [13], we have the following lemma.

Lemma 2.4

V maps bounded sets into bounded sets.
V is continuous.

In order to establish our multiplicity results, we recall an abstract critical point theorem, see [7,12]. Let X , Y be Banach spaces with X being separable and reflexive, and set E = X ⊕ Y . Let S ⊂ X ∗ be a countable dense subset. Let P be the family of semi-norms on E consisting of all semi-norms

p s : E = X ⊕ Y → R , p s ( x + y ) = ∣ s ( x ) ∣ + ‖ y ‖ , s ∈ S .

Denote by T P the topology on E induced by P . Let T w ∗ be the weak ∗ -topology of E ∗ .

For a functional Φ : E → R and numbers a , b ∈ R , we write Φ a ≔ { u ∈ E : Φ ( u ) ≤ a } , Φ a ≔ { u ∈ E : Φ ( u ) ≥ a } , and Φ a b ≔ Φ a ∩ Φ b . Assume

( Φ 1 ) Φ ∈ C 1 ( E , R ) ; Φ : ( E , T P ) → R is upper semi-continuous, and Φ ′ : ( Φ a , T P ) → ( E ∗ , T w ∗ ) is continuous for every a ∈ R ;

( Φ 2 ) There exists r > 0 with ρ ≔ inf Φ ( S r Y ) > Φ ( 0 ) = 0 , where S r Y ≔ { y ∈ Y : ‖ y ‖ = r } ;

( Φ 3 ) There exist a finite-dimensional subspace Y 0 ⊂ Y and R > r such that we have for E 0 ≔ X × Y 0 and B 0 ≔ { u ∈ E 0 : ‖ u ‖ ≤ R } , b ≔ sup Φ ( E 0 ) < ∞ and sup Φ ( E 0 \ B 0 ) < inf Φ ( B r Y ) .

We consider the set ℳ ( Φ c ) of maps g : Φ c → E with the properties

g is P -continuous and odd;
g ( Φ a ) ⊂ Φ a for all a ∈ [ ρ , b ] ;
Each u ∈ Φ c has a P -open neighborhood O ⊂ E such that the set ( i d − g ) ( O ∩ Φ c ) is contained in a finite-dimensional linear subspace.

The pseudo-index of Φ c is defined by

ψ ( c ) ≔ min { gen ( g ( Φ c ) ∩ S r Y ) : g ∈ ℳ ( Φ c ) } ∈ N 0 ∪ { ∞ } ,

where gen ( ⋅ ) denotes the usual symmetric index. Additionally, set for d > 0 fixed.

ℳ 0 ( Φ d ) ≔ { g ∈ ℳ ( Φ d ) : g is a homeomorphism from Φ d to g ( Φ d ) } .

Then we define for c ∈ [ 0 , d ]

ψ d ( c ) ≔ min { gen ( g ( Φ c ) ∩ S r Y ) : g ∈ ℳ 0 ( Φ d ) } .

Note that, by definition, ψ ( c ) ≤ ψ d ( c ) for all c ∈ [ 0 , d ] .

Theorem 2.1

[7,12] Let ( Φ 1 ) – ( Φ 3 ) be satisfied and assume that Φ is even and satisfies the ( C ) c -condition for c ∈ [ ρ , b ] . Then Φ has at least n ≔ dim Y 0 pairs of critical points with critical values given by

c i ≔ inf { c ≥ 0 : ψ ( c ) ≥ i } ∈ [ ρ , b ] , i = 1 , … , n .

If Φ has only finitely many critical points in Φ ρ b , then ρ < c 1 < c 2 < ⋯ < c n ≤ b .

Remark 2.1

Setting X = E − and Y = E + , it follows from the definition and Lemma 2.2 that the functional Φ = Φ ε is even and satisfies ( Φ 1 ) and ( Φ 2 ) . In the following, we are focusing on checking the assumption ( Φ 3 ) and the ( C ) c -condition.

3 Cut-off arguments

We will describe further the minimax value c ε by using a Mountain-Pass reduction technique, which depends on the convexity of the nonlinearities for verifying particularly that the second-order derivative of the functional is negative definite. However, the nonlocal term Γ λ ε destroys such a convexity, because its second-order derivative may take positive values (possibly very large) as well as negative values (possibly very large). In fact, the nonlocal power is 4 exceeding the growth of the nonlinear frequency term and dominates the behavior of the functional when ‖ u ‖ is very large. Therefore, we will adopt the cut-off argument of [13].

Denote T ≔ ( Λ + 1 ) 2 and let η : [ 0 , ∞ ) → [ 0 , 1 ] be a smooth function with η ( t ) = 1 if 0 ≤ t ≤ T , η ( t ) = 0 if t ≥ T + 1 , max ∣ η ′ ( t ) ∣ ≤ C 1 and max ∣ η ″ ( t ) ∣ ≤ C 2 . Without loss of generality, we can assume that η ( t ) is nonincreasing. Define

(3.1) Φ ˜ ε ( u ) = 1 2 ( ∥ u + ∥ 2 − ∥ u − ∥ 2 ) − η ( ‖ u ‖ 2 ) Γ λ ε ( u ) − Ψ ε ( u ) = 1 2 ( ∥ u + ∥ 2 − ∥ u − ∥ 2 ) − ℱ λ ε ( u ) − Ψ ε ( u ) .

By definition, Φ ε ∣ B T = Φ ˜ ε ∣ B T where B T ≔ { u ∈ E , ‖ u ‖ ≤ T } . Clearly,

(3.2) 0 ≤ ℱ λ ε ( u ) ≤ Γ λ ε ( u )

and

ℱ λ ε ′ ( u ) v = 2 η ′ ( ‖ u ‖ 2 ) Γ λ ε ( u ) ⟨ u , v ⟩ + η ( ‖ u ‖ 2 ) Γ λ ε ′ ( u ) v for u , v ∈ E .

3.1 The limit equation: subcritical case

For any 0 < μ ≤ κ , 0 < λ ≤ λ ¯ , we consider the following constant coefficient system:

(3.3) i α ⋅ ∇ u − a β u + ω u − λ V β u = μ ∣ u ∣ p − 2 u , − Δ V + M ⋅ V = 4 π λ ( β u ) u .

As before, we consider the modified functional

(3.4) ϕ λ , μ ( u ) = 1 2 ( ∥ u + ∥ 2 − ∥ u − ∥ 2 ) − η ( ‖ u ‖ 2 ) Γ λ ( u ) − Ψ μ ( u ) = 1 2 ( ∥ u + ∥ 2 − ∥ u − ∥ 2 ) − ℱ λ ( u ) − μ p ∫ R 3 ∣ u ∣ p d x ,

defined for u = u + + u − ∈ E . Obviously, ϕ λ , μ possesses the linking structure. By γ λ , μ we denote the linking level of ϕ λ , μ . Define ℓ λ , μ : E + → E − by, for u ∈ E + ,

ϕ λ , μ ( u + ℓ λ , μ ( u ) ) = max v ∈ E − ϕ λ , μ ( u + v )

and I λ , μ ( u ) = ϕ λ , μ ( u + ℓ λ , μ ( u ) ) . Then I λ , μ ∈ C 2 ( E + , R ) and u ∈ E + is a critical point of I λ , μ if and only if u + ℓ λ , μ ( u ) is a critical point of ϕ λ , μ . Set ℳ λ , μ = { u ∈ E + : I λ , μ ′ ( u ) u = 0 } . We will call ( ℓ λ , μ ( ⋅ ) , I λ , μ ( ⋅ ) , ℳ λ , μ ) the Mountain-Pass reduction of (3.1). It is known that

ℒ λ , μ ≔ { u ∈ E : ϕ λ , μ ′ ( u ) = 0 } ≠ ∅ and ℒ λ , μ ⊂ ⋂ q ≥ 2 W 1 , q ;
γ λ , μ ≔ inf { ϕ λ , μ ( u ) : u ∈ ℒ λ , μ \ { 0 } } = inf u ∈ ℳ λ , μ I λ , μ ( u ) > 0 and is attained;
ℛ λ , μ ≔ { u ∈ ℒ λ , μ : ϕ λ , μ ( u ) = γ λ , μ , ∣ u ( 0 ) ∣ = ∣ u ∣ ∞ } is compact in H 1 , and there exist C , c > 0 such that ∣ u ( x ) ∣ ≤ C exp ( − c ∣ x ∣ ) for all x ∈ R 3 and u ∈ ℛ λ , μ .

Lemma 3.1

Let γ ∞ denote the least energy of (3.4) with λ = λ ∞ , μ = κ ∞ . For any λ ∞ < λ ≤ λ ¯ , κ ∞ < μ ≤ κ , there holds m ( λ , μ ) γ λ , μ < γ ∞ , where m ( λ , μ ) ≔ min λ λ ∞ 2 , μ κ ∞ 2 p − 2 .

Proof

Let u be a least energy critical point of

ϕ λ ∞ , μ ∞ ( u ) = 1 2 ( ∥ u + ∥ 2 − ∥ u − ∥ 2 ) − η ( ‖ u ‖ 2 ) Γ λ ∞ ( u ) − Ψ κ ∞ ( u ) = 1 2 ( ∥ u + ∥ 2 − ∥ u − ∥ 2 ) − ℱ λ ∞ ( u ) − μ ∞ p ∫ R 3 ∣ u ∣ p d x ,

with the energy γ ∞ . For any λ ∞ < λ ≤ λ ¯ , κ ∞ < μ ≤ κ , set

v ( x ) = b u ( x ) with 1 > b ≥ max λ ∞ λ , κ ∞ μ 1 p − 2 .

Then

γ ∞ = 1 2 b 2 ( ∥ v + ∥ 2 − ∥ v − ∥ 2 ) − η 1 b 2 ‖ v ‖ 2 b 4 Γ λ ∞ ( v ) − 1 b p Ψ κ ∞ ( v ) > 1 2 b 2 ( ∥ v + ∥ 2 − ∥ v − ∥ 2 ) − η ( ‖ v ‖ 2 ) b 4 Γ λ ∞ ( v ) − 1 b p Ψ κ ∞ ( v ) ≥ 1 b 2 1 2 ( ∥ v + ∥ 2 − ∥ v − ∥ 2 ) − η ( ‖ v ‖ 2 ) Γ λ ( v ) − Ψ κ ( v ) ≥ 1 b 2 γ λ , μ .

Here we use the fact that λ ∞ b λ ≤ 1 and κ ∞ b p − 2 μ ≤ 1 . Thus, m ( λ , μ ) γ λ , μ < γ ∞ .□

3.2 The limit equation: critical case

For any 0 < λ ≤ λ ¯ , 0 < μ j ≤ κ j , j = 1 , 2 , we consider the following system:

(3.5) i α ⋅ ∇ u − a β u + ω u − λ V β u = μ 1 ∣ u ∣ p − 2 u + μ 2 ∣ u ∣ u , − Δ V + M ⋅ V = 4 π λ ( β u ) u .

As before, we consider the modified functional

(3.6) ϕ λ , μ → ( u ) ≔ 1 2 ( ∥ u + ∥ 2 − ∥ u − ∥ 2 ) − η ( ‖ u ‖ 2 ) Γ λ ( u ) − Ψ μ ( u ) = 1 2 ( ∥ u + ∥ 2 − ∥ u − ∥ 2 ) − ℱ λ ( u ) − μ 1 p ∫ R 3 ∣ u ∣ p d x − μ 2 3 ∫ R 3 ∣ u ∣ 3 d x ,

where μ → ≔ ( μ 1 , μ 2 ) . As above, let γ λ , μ → denote the linking level of ϕ λ , μ → . Define ℓ λ , μ → : E + → E − by, for u ∈ E + ,

ϕ λ , μ → ( u + ℓ λ , μ → ( u ) ) = max v ∈ E − ϕ λ , μ → ( u + v )

and I λ , μ → ( u ) = ϕ λ , μ → ( u + ℓ λ , μ → ( u ) ) . Then I λ , μ → ∈ C 2 ( E + , R ) and u ∈ E + is a critical point of I λ , μ → if and only if u + ℓ λ , μ → ( u ) is a critical point of ϕ λ , μ → . Set ℳ λ , μ → = { u ∈ E + : I λ , μ → ′ ( u ) u = 0 } . Write as before ℒ λ , μ → and ℛ λ , μ → . One has γ λ , μ → ≔ inf { ϕ λ , μ → ( u ) : u ∈ ℒ λ , μ → \ { 0 } } = inf u ∈ ℳ λ , μ → I λ , μ → ( u ) > 0 and ℒ λ , μ → ⊂ ⋂ q ≥ 2 W 1 , q .

Lemma 3.2

γ λ , μ → is attained provided γ λ , μ → < S 3 2 6 μ 2 2 .

Proof

Let { u n } be a ( C ) c -sequence with c = γ λ , μ → . By the statements in Corollary 2.1, { u n } is bounded in E . By Lion’s concentration principle [27], { u n } is either vanishing or nonvanishing.

Assume that { u n } is vanishing. Then ∣ u n ∣ s → 0 for s ∈ ( 2 , 3 ) , together with Lemma 2.1 implies that ℱ λ ( u n ) ≤ S − 1 λ ¯ 2 ∣ u n ∣ 12 5 4 → 0 . Thus, one obtains

γ λ , μ → + o ( 1 ) = ϕ λ , μ → ( u n ) − 1 2 ϕ λ , μ → ′ ( u n ) u n = ℱ λ ( u n ) + ∫ R 3 1 2 − 1 6 μ 1 ∣ u n ∣ p d x + ∫ R 3 1 2 − 1 3 μ 2 ∣ u n ∣ 3 d x = ∫ R 3 1 6 μ 2 ∣ u n ∣ 3 d x + o ( 1 ) ,

that is, ∫ R 3 ∣ u n ∣ 3 d x = 6 γ λ , μ → μ 2 + o ( 1 ) . Similarly, we also have

o ( 1 ) = ϕ λ , μ → ′ ( u n ) ( u n + − u n − ) = ‖ u n ‖ 2 − ℱ λ ′ ( u n ) ( u n + − u n − ) − ℜ ∫ R 3 μ 1 ∣ u n ∣ p − 2 u n ( u n + − u n − ) d x − ℜ ∫ R 3 μ 2 ∣ u n ∣ u n ( u n + − u n − ) d x = ‖ u n ‖ 2 − ℜ ∫ R 3 μ 2 ∣ u n ∣ u n ( u n + − u n − ) d x + o ( 1 ) .

Thus, jointly with the fact S 1 2 ∣ u ∣ 3 2 ≤ ‖ u ‖ 2 [8], we have

‖ u n ‖ 2 ≤ μ 2 ∣ u n ∣ 3 ∣ u n ∣ 3 ∣ u n + − u n − ∣ 3 + o ( 1 ) ≤ μ 2 S − 1 2 ‖ u n ‖ 6 γ λ , μ → μ 2 1 3 ‖ u n + − u n − ‖ + o ( 1 ) ,

which implies γ λ , μ → ≥ S 3 2 6 μ 2 2 , a contradiction.

Therefore, { u n } is nonvanishing, that is, there exist r , δ > 0 and x n ∈ R 3 such that, setting v n ( x ) = u n ( x + x n ) , along a subsequence,

∫ B r ( 0 ) ∣ v n ∣ 2 d x ≥ δ .

Without loss of generality, we assume v n → v . Then v ≠ 0 and is a solution of (3.5). And so γ λ , μ → is attained.□

Lemma 3.3

γ λ , μ → is attained, provided μ 1 − 1 μ 2 p − 2 < S 3 2 6 τ p p − 2 2 .

Proof

By the reduction process and the minmax scheme, we deduce

γ λ , μ → ≤ γ μ 1 = μ 1 − 2 p − 2 τ p .

If μ 1 − 1 μ 2 p − 2 < S 3 2 6 τ p p − 2 2 , then γ λ , μ → ≤ μ 1 − 2 p − 2 τ p ≤ S 3 2 6 μ 2 2 . So γ λ , μ → is attained by Lemma 3.2.□

In the sequel, by μ → 1 < μ → 2 we mean that min { μ 1 2 − μ 1 1 , μ 2 2 − μ 2 1 } > 0 for any vectors μ → j = ( μ 1 j , μ 2 j ) . The following lemma will be useful to study our problem.

Lemma 3.4

Let u ∈ ℳ λ , μ → be such that I λ , μ → ( u ) = γ λ , μ → and set E u = E − ⊕ R + u . Then
max w ∈ E u ϕ λ , μ → ( w ) = I λ , μ → ( u ) .
If λ 2 < λ 1 , μ → 1 − μ → 2 > 0 , then γ λ 1 , μ → 1 < γ λ 2 , μ → 2 .

Proof

To prove (1), we note that u + ℓ λ , μ → ( u ) ∈ E u and

I λ , μ → ( u ) = ϕ λ , μ → ( u + ℓ λ , μ → ( u ) ) ≤ max w ∈ E u ϕ λ , μ → ( w ) .

Moreover, since u ∈ ℳ λ , μ → ,

max w ∈ E u ϕ λ , μ → ( w ) ≤ max s ≥ 0 ϕ λ , μ → ( s u + ℓ λ , μ → ( s u ) ) ≤ max s ≥ 0 I λ , μ → ( s u ) = I λ , μ → ( u ) .

Therefore, max w ∈ E u ϕ λ , μ → ( w ) = I λ , μ → ( u ) .

To obtain (2), let u 2 be a least energy solution of ϕ λ 2 , μ → 2 and set e = u 2 + . Then

γ λ 2 , μ → 2 = ϕ λ 2 , μ → 2 ( u 2 ) = max w ∈ E e ϕ λ 2 , μ → 2 ( w ) .

Suppose u 1 ∈ E e be such that ϕ λ 1 , μ → 1 ( u 1 ) = max w ∈ E e ϕ λ 1 , μ → 1 ( w ) . We deduce that

γ λ 2 , μ → 2 = ϕ λ 2 , μ → 2 ( u 2 ) ≥ ϕ λ 2 , μ → 2 ( u 1 ) > ϕ λ 1 , μ → 1 ( u 1 ) .

This ends the proof.□

Lemma 3.5

Let γ ∞ → denote the least energy of (3.5) with λ = λ ∞ , μ → = κ ∞ → with κ ∞ → ≔ ( κ 1 ∞ , κ 1 ∞ ) . For any λ ∞ < λ ≤ λ ¯ , κ j ∞ < μ j ≤ κ j , there holds

m ( λ , μ → ) γ λ , μ → < γ ∞ → , where m ( λ , μ → ) ≔ min λ λ ∞ 2 , μ 1 κ 1 ∞ 2 p − 2 , μ 2 κ 2 ∞ 2 .

Proof

Let u be a least energy critical point of

ϕ λ ∞ , κ ∞ → ( u ) = 1 2 ( ∥ u + ∥ 2 − ∥ u − ∥ 2 ) − η ( ‖ u ‖ 2 ) Γ λ ∞ ( u ) − Ψ κ ∞ → ( u ) = 1 2 ( ∥ u + ∥ 2 − ∥ u − ∥ 2 ) − ℱ λ ∞ ( u ) − μ 1 ∞ p ∫ R 3 ∣ u ∣ p d x − μ 2 ∞ 3 ∫ R 3 ∣ u ∣ 3 d x ,

with the energy γ ∞ → . For any λ ∞ < λ ≤ λ ¯ , κ ∞ < μ ≤ κ , set

v ( x ) = b u ( x ) with 1 > b ≥ max λ λ ∞ , μ 1 κ 1 ∞ 1 p − 2 , μ 2 κ 2 ∞ .

Then

γ ∞ → = 1 2 b 2 ( ∥ v + ∥ 2 − ∥ v − ∥ 2 ) − η 1 b 2 ‖ v ‖ 2 b 4 Γ λ ∞ ( v ) − 1 b p Ψ κ ∞ → ( v ) > 1 2 b 2 ( ∥ v + ∥ 2 − ∥ v − ∥ 2 ) − η ( ‖ v ‖ 2 ) b 4 Γ λ ∞ ( v ) − 1 b p κ 1 ∞ p ∫ R 3 ∣ v ∣ p d x − 1 b 3 κ 2 ∞ 3 ∫ R 3 ∣ v ∣ 3 d x ≥ 1 b 2 1 2 ( ∥ v + ∥ 2 − ∥ v − ∥ 2 ) − η ( ‖ v ‖ 2 ) Γ λ ( v ) − Ψ μ → ( v ) ≥ 1 b 2 γ λ , μ → .

Here we use the fact that λ ∞ b λ ≤ 1 , κ 1 ∞ b p − 2 μ 1 ≤ 1 , and κ 2 ∞ b μ 2 ≤ 1 . Thus, m ( λ , μ ) γ λ , μ < γ ∞ → .□

3.3 The cut-off functionals

Let λ ˆ = inf λ ( x ) , b = inf W ( x ) , and b → = ( b 1 , b 2 ) with b j = inf W j ( x ) . Take

e 0 ∈ ℛ λ ˆ , b , for the subcritical case , ℛ λ ˆ , b → , for the critical case ,

and set

c λ ˆ , b → = γ λ ˆ , b , for the subcritical case , γ λ ˆ , b → , for the critical case , c ∞ = γ ∞ , for the subcritical case , γ ∞ → , for the critical case .

Clearly, c ε ≤ c λ ˆ , b → for all ε > 0 and c ∞ ≤ c λ ˆ , b → . In addition, one has

Lemma 3.6

For all ε > 0 , max w ∈ E e 0 Φ ˜ ε ( w ) ≤ c λ ˆ , b → .

Proof

It is clear that Φ ˜ ε ( u ) ≤ ϕ λ ˆ , b → for all u ∈ E , hence, by Lemma 3.4(1)

max w ∈ E e 0 Φ ˜ ε ( w ) ≤ max w ∈ E e 0 ϕ λ ˆ , b ( w ) = I λ ˆ , b → ( e 0 ) = c λ ˆ , b →

as claimed.□

Lemma 3.7

There exists ε 1 > 0 and λ 0 > 0 such that, for any ε ∈ ( 0 , ε 1 ) and λ ¯ ∈ ( 0 , λ 0 ) , if { u n ε } is a (C) c sequence of Φ ˜ ε , then ∥ u n ε ∥ ≤ Λ + 1 2 , and consequently Φ ˜ ε ( u n ε ) = Φ ε ( u n ε ) .

In particular, replace Φ ˜ ε with ϕ ˜ λ , μ → , we obtain that ϕ ˜ λ , μ → shares the same ground state solution with ϕ λ , μ → .

Proof

We repeat the arguments of Lemma 2.3. If ‖ u ‖ 2 ≥ T + 1 , then ℱ λ ε ( u ) = 0 . Thus, as proved in Lemma 2.3, one obtains ‖ u ‖ 2 ≤ Λ , a contradiction. We assume that ‖ u ‖ 2 ≤ T + 1 . Then, using Lemma 2.1, ∣ η ′ ( ‖ u ‖ 2 ) ‖ u ‖ 2 Γ λ ε ( u ) ∣ ≤ λ ¯ 2 d λ ( 1 ) (here and in the following, by d λ ( j ) we denote positive constants depending only on λ and d λ ( j ) is increasing with respect to λ ). Similar to (2.10),

c 1 + 1 ≥ Φ ˜ ε ( u ) − 1 2 Φ ˜ ε ′ ( u ) u = η ( ‖ u ‖ 2 ) + 2 η ′ ( ‖ u ‖ 2 ) ‖ u ‖ 2 Γ λ ε ( u ) − 1 p − 1 2 ∫ R 3 W 1 ( ε x ) ∣ u ∣ p d x + 1 6 ∫ R 3 W 2 ( ε x ) ∣ u ∣ 3 d x ≥ − λ ¯ 2 d λ ( 1 ) + 1 2 − 1 p inf W 1 ∣ u ∣ p p + inf W 2 6 ∣ u ∣ 3 3 ,

that is,

∣ u ∣ 3 ≤ d λ ( 2 ) and ∣ u ∣ p ≤ d λ ( 3 ) .

Similarly, we obtain that

c 2 ≥ Φ ˜ ε ′ ( u ) ( u + − u − ) ≥ ‖ u ‖ 2 − λ ¯ 2 d λ ( 1 ) − λ ¯ 2 C 4 ‖ u ‖ 1 + ξ − κ 1 ∣ u ∣ p p − κ 2 ∣ u ∣ 3 3 ≥ ‖ u ‖ 2 − λ ¯ 2 d λ ( 1 ) − λ ¯ 2 C 4 ‖ u ‖ 1 + ξ − κ 1 ( d λ ( 2 ) ) p − κ 2 ( d λ ( 3 ) ) 3 .

That is,

‖ u ‖ 2 ≤ λ ¯ 2 C 4 ‖ u ‖ 1 + ξ + λ ¯ 2 d λ ( 1 ) + d λ ( 4 ) .

By monotonicity of d λ ( j ) , we see that, for λ 0 > 0 being suitably chosen, let λ ¯ ∈ ( 0 , λ 0 ] then ‖ u ‖ ≤ Λ + 1 2 . The proof is complete.□

Based on this lemma, to prove our main results, it suffices to study Φ ˜ ε and obtain its critical points with critical values in [ 0 , c ˜ λ ˆ , b → ] . This will be done via a series of arguments. The first is to introduce the minmax values of Φ ˜ ε . It is easy to verify the following lemma by a similar argument of Lemma 2.2.

Lemma 3.8

Φ ˜ ε possesses a linking structure and the constants in Lemma 2.2 are true for Φ ˜ ε . In addition, max v ∈ E e 0 Φ ˜ ε ( v ) ≤ c ˜ λ ˆ , b → .

Let c ˜ ε denote the minimax level of Φ ˜ ε induced by the linking structure defined by (2.8) with Φ ε replaced by Φ ˜ ε . By (3.6) and the forms of the functionals, one sees c ε ≤ c ˜ ε . As before, define h ε : E + → E − by

Φ ˜ ε ( u + h ε ( u ) ) = max v ∈ E − Φ ˜ ε ( u + v ) ,

and h λ , μ → : E + → E − by

ϕ ˜ λ , μ → ( u + h λ , μ → ( u ) ) = max v ∈ E − ϕ ˜ λ , μ → ( u + v ) .

The following is known (see [4])

h ε , h λ , μ → ∈ C 1 ( E + , E − ) , h ε ( 0 ) = 0 , h λ , μ → ( 0 ) = 0 ;
h ε , h λ , μ → are bounded maps;
If u n ⇀ u in E + , then
h ε ( u n ) − h ε ( u n − u ) → h ε ( u ) and h ε ( u n ) ⇀ h ε ( u )

h λ , μ → ( u n ) − h λ , μ → ( u n − u ) → h λ , μ → ( u ) and h λ , μ → ( u n ) ⇀ h λ , μ → ( u ) .
We now define I ε : E + → R by I ε ( u ) = Φ ˜ ε ( u + h ε ( u ) ) , I λ , μ → : E + → R by I λ , μ → ( u ) = Φ ˜ λ , μ → ( u + h λ , μ → ( u ) ) , and set

(3.7) N ε ≔ { u ∈ E + \ { 0 } : I ε ′ ( u ) u = 0 } , N λ , μ → ≔ { u ∈ E + \ { 0 } : I λ , μ → ′ ( u ) u = 0 } .

Clearly,

(3.8) c ˜ ε = inf u ∈ N ε I ε ( u ) ≤ c ˜ λ ˆ , b .

This, jointly with (2.8), implies that there is a sequence { e n } ⊂ E + \ { 0 } such that, denoting u n = e n + h ε ( e n ) , Φ ˜ ε ( u n ) → c ˜ ε and Φ ˜ ε ′ ( u n ) → 0 as n → ∞ . Consequently, by Lemma 3.7 one has

c ˜ ε = c ε for all ε > 0 small .

From now on we will only use c ε .

Recall that λ ( ε x ) → λ ¯ , W ( ε x ) → κ , and ( W 1 ( ε x ) , W 2 ( ε x ) ) → ( κ 1 , κ 2 ) = κ → as ε → 0 uniformly in bounded sets of x .

By a similar argument to [4,13,15], one has

Lemma 3.9

For any u ∈ E + \ { 0 } , there is a unique t ε = t ε ( u ) > 0 such that t ε u ∈ N ε . Moreover, { t ε ( u ) } ε ≤ 1 is bounded, and if along a subsequence t ε ( u ) → t 0 ( u ) , then ‖ h ε ( t ε u ) − h λ ¯ , κ → ( t 0 u ) ‖ → 0 . If u ∈ N λ ¯ , κ → , then t 0 = 1 .

In addition, we have

Lemma 3.10

lim ε → 0 c ˜ ε = γ ˜ λ ¯ , κ → .

Proof

First, we show that liminf c ε ≥ γ ˜ λ ¯ , κ → . Arguing indirectly, assume that liminf c ε < γ ˜ λ ¯ , κ → . By the definition of c ε and (3.8), we can choose an e j ∈ N ε and δ > 0 such that max u ∈ E e j Φ ˜ ε j ( u ) ≤ γ ˜ λ ¯ , κ → − δ , as ε j → 0 . Clearly, Φ ˜ ε j ( u ) ≥ ϕ ˜ λ ¯ , κ → ( u ) for all u ∈ E and ε small. Note also that γ ˜ λ ¯ , κ → ≤ I λ ¯ , κ → ( e j ) ≤ max u ∈ E e j ϕ ˜ λ ¯ , κ → ( u ) . Therefore, we obtain, for all ε j small,

γ ˜ λ ¯ , κ → − δ ≥ max u ∈ E e j Φ ˜ ε j ( u ) ≥ max u ∈ E e j ϕ ˜ λ ¯ , κ → ( u ) ≥ γ ˜ λ ¯ , κ → ,

a contradiction.□

Next take λ ∞ < λ < λ ¯ , κ ∞ < μ < κ , κ 1 ∞ < μ 1 < κ 1 and define

λ λ ( x ) = min { λ , λ ( x ) } , W μ ( x ) = min { μ , W ( x ) } , W 1 μ 1 ( x ) = min { μ 1 , W 1 ( x ) } .

We consider the truncated energy functional

Φ ˜ ε λ , μ = 1 2 ( ∥ u + ∥ 2 − ∥ u − ∥ 2 ) − ℱ ε λ ( u ) − Ψ ε μ ( u ) ,

where

Ψ ε μ = ∫ R 3 1 p W ε μ ( x ) ∣ u ∣ p d x in the subcritical case ∫ R 3 1 p W 1 ε μ 1 ( x ) ∣ u ∣ p d x + ∫ R 3 1 3 W 2 ( ε x ) ∣ u ∣ 3 d x in the critical case ,

and ℱ ε λ = 1 4 η ( ‖ u ‖ 2 ) Γ λ ε λ , λ ε λ ( x ) = λ λ ( ε x ) , W ε μ ( x ) = W μ ( ε x ) , W 1 ε 1 μ ( x ) = W 1 μ ( ε x ) . As before define correspondingly h ˜ ε λ , μ : E + → E − , I ˜ ε λ , μ : E + → R , N ˜ ε λ , μ , c ˜ ε λ , μ , and so on.

Lemma 3.11

c ˜ ε is attained for small ε > 0 .

Proof

Given ε > 0 , let { u n } ⊂ N ε be a minimization sequence: I ε ( u n ) → c ε . By the Ekeland variational principle, we can assume that { u n } is in fact a ( P S ) c ε -sequence for I ε on E + . Then w n = u n + h ε ( u n ) is a ( P S ) c ε -sequence for Φ ˜ ε on E . It is clear that { w n } is bounded, hence is a ( C ) c ε -sequence. We can assume without loss of generality that w n ⇀ w ε in E . If w ε ≠ 0 , it is easy to check that Φ ˜ ε ( w ε ) = c ε . So we are going to show that w ε ≠ 0 for all small ε > 0 . Assume by contradiction that there is a sequence ε j → 0 with w ε j = 0 , then w n = u n + h ε j ( u n ) ⇀ 0 in E , u n → 0 in L loc q for q ∈ [ 1 , 3 ) , and w n ( x ) → 0 a.e. in x ∈ R 3 . Let t n > 0 be such that t n u n ∈ N ˜ ε λ , μ . Since u n ∈ N ε , it is not difficult to see { t n } is bounded and one may assume t n → t 0 as n → ∞ . Remark that h ˜ ε λ , μ ( t n u n ) ⇀ 0 in E and h ˜ ε λ , μ ( t n u n ) ⇀ 0 in L loc q for q ∈ [ 1 , 3 ) . w ˜ n ≔ t n u n + h ˜ ε λ , μ ( t n u n ) → 0 in L loc q for q ∈ [ 1 , 3 ) . Set A ε ≔ { x ∈ R 3 : λ ( ε x ) > λ } is bounded. Thus,

∣ ℱ ε j λ − ℱ ε j ∣ ≤ C ∫ R 3 ∫ R 3 ( λ ε j λ ( x ) λ ε j λ ( y ) − λ ε j ( x ) λ ε j ( y ) ) [ ( β w ˜ n ) w ˜ n ] ( x ) [ ( β w ˜ n ) w ˜ n ] ( y ) ∣ x − y ∣ e − M ∣ x − y ∣ d y d x = ∫ A ε ∫ A ε c ( λ − λ ε j ( x ) ) λ [ ( β w ˜ n ) w ˜ n ] ( x ) [ ( β w ˜ n ) w ˜ n ] ( y ) ∣ x − y ∣ e − M ∣ x − y ∣ d y d x + ∫ A ε c ∫ A ε ( λ − λ ε j ( y ) ) λ [ ( β w ˜ n ) w ˜ n ] ( x ) [ ( β w ˜ n ) w ˜ n ] ( y ) ∣ x − y ∣ e − M ∣ x − y ∣ d y d x + ∫ A ε ∫ A ε ( λ λ − λ ε j ( x ) λ ε j ( y ) ) [ ( β w ˜ n ) w ˜ n ] ( x ) [ ( β w ˜ n ) w ˜ n ] ( y ) ∣ x − y ∣ e − M ∣ x − y ∣ d y d x ≤ C ∫ A ε ∣ w ˜ n ∣ 12 5 5 6 ∫ R 3 ∣ V w ˜ n ∣ 6 1 6 + ∫ A ε ∣ w ˜ n ∣ 12 5 5 6 ≤ C ∫ A ε ∣ w ˜ n ∣ 12 5 5 6 ∫ R 3 ∣ w ˜ n ∣ 12 5 5 6 + C ∫ A ε ∣ w ˜ n ∣ 12 5 5 6 → 0 ,

where V w ˜ n ( x ) = ∫ R 3 [ ( β w ˜ n ) w ˜ n ] ( y ) ∣ x − y ∣ e − M ∣ x − y ∣ d y .

Similarly, since set { x ∈ R 3 : W 1 ( ε x ) > μ } is bounded, we have

∫ R 3 1 p W 1 ε μ 1 ( x ) ∣ w ˜ n ∣ p d x − ∫ R 3 1 p W 1 ε ( x ) ∣ w ˜ n ∣ p d x = o ( 1 ) .

Therefore, we obtain

c ˜ ε j λ , μ ≤ I ˜ ε j λ , μ ( t n u n ) = Φ ˜ ε j λ , μ ( w ˜ n ) = Φ ˜ ε j ( w ˜ n ) − ℱ ε j λ + ℱ ε j − ∫ R 3 1 p W 1 ε μ 1 ( x ) ∣ w ˜ n ∣ p d x + ∫ R 3 1 p W 1 ε ( x ) ∣ w ˜ n ∣ p d x ≤ I ˜ ε j ( u n ) + o ( 1 ) = c ˜ ε j + o ( 1 ) .

For the subcritical case, as done in the proof of Lemma 3.10, lim ε → 0 c ˜ ε j λ , μ = γ ˜ λ , μ . Thus, γ ˜ λ , μ ≤ γ ˜ λ ¯ , κ , which contradicts with γ ˜ λ , μ > γ ˜ λ ¯ , κ .

For the critical case, note that γ ˜ λ , μ → ≤ c ˜ ε j λ , μ ≤ c ˜ ε j , where μ → ≔ ( μ 1 , κ 2 ) . Thus, by Lemma 3.10, we have γ ˜ λ , μ → ≤ γ ˜ λ ¯ , κ → , which contradicts with γ ˜ λ , μ → > γ ˜ λ ¯ , κ → .□

We now turn to prove the desired conclusion.

Lemma 3.12

Let u n = u n + + u n − be a ( C ) c -sequence for Φ ˜ ε and set v n = u n + + h ε ( u n + ) z n = u n − − h ε ( u n + ) . Then ∥ z n ∥ → 0 and { v n } is also a ( C ) c -sequence for Φ ˜ ε , that is, { u n + } is a ( C ) c -sequence for I ε . Consequently, either c = 0 or c ≥ c ε .

Proof

It suffices to show that ∥ z n ∥ → 0 . Note that, by Lemma 3.7, one has ∥ u n ∥ ≤ Λ + 1 2 , hence { u n } is in fact a bounded ( C ) c sequence for Φ ε : Φ ε ( u n ) → c and Φ ε ′ ( u n ) → 0 . Observe that

0 = Φ ε ′ ( v n ) z n = − ( h ε ( u n + ) , z n ) − Γ λ ε ′ ( v n ) z n − Ψ ε ′ ( v n ) z n

and

o ( 1 ) = Φ ε ′ ( u n ) z n = − ( u n − , z n ) − Γ λ ε ′ ( u n ) z n − Ψ ε ′ ( u n ) z n .

Following [13, Remark 3.10], one has

∣ Γ λ ε ″ ( v n + t z n ) [ z n , z n ] ∣ ≤ 1 2 ∥ z n ∥ 2 .

Thus,

o ( 1 ) = ∥ z n ∥ 2 + ( Γ λ ε ′ ( v n + z n ) − Γ λ ε ′ ( v n ) ) z n + ( Ψ ε ′ ( v n + z n ) − Ψ ε ′ ( v n ) ) z n ≥ ∥ z n ∥ 2 ,

which shows ∥ z n ∥ → 0 . Finally, it follows from (2.8) that if c ≠ 0 then c ≥ c ε .□

We turn to study the ( C ) c -condition of Φ ˜ .

Lemma 3.13

For all ε > 0 small, Φ ˜ ε satisfies the ( C ) c condition for all c < c ∞ .

Proof

For later aims we set

λ ∞ ( x ) = min { λ ∞ , λ ( x ) } , W ∞ ( x ) = min { κ ∞ , W ( x ) } , W j ∞ ( x ) = min { κ j ∞ , W j ( x ) } , j = 1 , 2 .

Define

Ψ ε ∞ = ∫ R 3 1 p W ε ∞ ( x ) ∣ u ∣ p d x , in the subcritical case , ∫ R 3 1 p W 1 ε ∞ ( x ) ∣ u ∣ p d x + ∫ R 3 1 3 W 2 ε ∞ ( x ) ∣ u ∣ 3 d x , in the critical case ,

ℱ ε ∞ = 1 4 η ( ‖ u ‖ 2 ) Γ λ ε ∞ ,

and

(3.9) Φ ˜ ε ∞ = 1 2 ( ∥ u + ∥ 2 − ∥ u − ∥ 2 ) − ℱ ε ∞ ( u ) − Ψ ε ∞ ( u ) ,

where λ ε ∞ ( x ) = λ ∞ ( ε x ) , W ε ∞ ( x ) = W ∞ ( ε x ) , W j ε ∞ ( x ) = W j ∞ ( ε x ) . It is not difficult to verify that Φ ˜ ε ∞ has the same properties possessed by Φ ˜ ε shown above. In particular, letting c ε ∞ be the linking level of Φ ˜ ε ∞ , we have c ε ∞ → c ∞ as ε → 0 .

Let { u n } be a ( C ) c -sequence for Φ ˜ ε with c < c ∞ . By virtue of Lemma 3.2, { u n } is bounded and Φ ˜ ε ( u n ) = Φ ε ( u n ) , hence it is a ( C ) c -sequence for Φ ε . We can assume that u n ⇀ u . Clearly, Φ ε ′ ( u ) = 0 . Set z n = u n − u . Note that z n ⇀ 0 in E , z n → 0 in L loc q ∈ [ 1 , 3 ) , and z n ( x ) → 0 a.e. in x . Using the Brezis-Lieb lemma [33]:

∫ R 3 ∣ u n ∣ q d x − ∫ R 3 ∣ z n ∣ q d x → ∫ R 3 ∣ u ∣ q d x ,

it is easy to check that Φ ε ∞ ( z n ) → c − Φ ε ( u ) and ( Φ ε ∞ ) ′ ( z n ) → 0 . If c = Φ ε ( u ) , then z n → 0 and we are done. If c − Φ ε ( u ) ≥ c ε ∞ , then c ≥ c ε + c ε ∞ , a contradiction.□

Let K ε ≔ { u ∈ E : Φ ˜ ε ′ ( u ) = 0 } be the critical set of Φ ˜ ε . By using the same iterative argument of [22] one obtains easily the following (see [13,16]).

Lemma 3.14

If u ∈ K ε with ∣ Φ ˜ ε ( u ) ∣ ≤ C 1 , then, for any q ∈ [ 2 , + ∞ ) , ‖ u ‖ W 1 , q ≤ Λ q where Λ q depends only on C 1 .

4 Proof of main results

4.1 Proof of main results: the subcritical case

Without loss of generality, we may assume that 0 ∈ W ∩ Γ , hence, λ ¯ = λ ( 0 ) , κ = W ( 0 ) . Solutions of (2.7) are critical points of the functional Φ ε ( u ) = Φ ε λ ¯ , κ ( u ) . For notational convenience we denote Φ 0 ( u ) = ϕ λ ¯ , κ ( u ) . We will utilize Theorem 2.1. Obviously, Φ ε is even, and in virtue of Remark 2.1 the conditions ( Φ 1 ) and ( Φ 2 ) are satisfied. It remains to verify ( Φ 3 ) .

Let u ∈ ℛ λ ¯ , κ and let χ r ∈ C 0 ∞ ( R + ) be such that χ r ( s ) = 1 if s ≤ r , χ r ( s ) = 0 if s ≥ r + 1 , and χ r ′ ( s ) ≤ 0 . Set u r ( x ) = χ r ( ∣ x ∣ ) u ( x ) . Recall that ∣ u ( x ) ∣ ≤ C e − c ∣ x ∣ for some C , c > 0 and all x ∈ R 3 , hence ∥ u r − u ∥ → 0 as r → ∞ . Then ∥ u r ± − u ± ∥ ≤ ∥ u r − u ∥ → 0 , Φ 0 ( u r ) → γ λ ¯ , κ and Φ 0 ′ ( u r ) → 0 , as r → ∞ . Let h 0 : E + → E − be defined so that Φ 0 ( u + h 0 ( u ) ) = max v ∈ E − Φ 0 ( u + v ) . Clearly, ∥ u r − − h 0 ( u r + ) ∥ → 0 , and ∥ u r − u ˆ r ∥ → 0 , where u ˆ r = u r + + h 0 ( u r + ) (see Lemma 3.12). Therefore,

(4.1) max v ∈ E − Φ 0 ( u r + + v ) = Φ 0 ( u ˆ r ) = Φ 0 ( u r ) + o ( 1 ) = γ λ ¯ , κ + o ( 1 ) ,

as ε → 0 . Observe that since, as ε → 0 , λ ε → λ ¯ and W ( ε x ) → κ uniformly in ∣ x ∣ ≤ r + 1 we have that, for any δ > 0 , there are r δ > 0 and ε δ > 0 such that

(4.2) max w ∈ E − ⊕ R u r Φ ε ( w ) < γ λ ¯ , κ + δ ,

for all r ≥ r δ and ε ≤ ε δ .

Let y j = ( 2 j ( r + 1 ) , 0 , 0 ) , define u j ( x ) = u ( x − y j ) = u ( x 1 − 2 j ( r + 1 ) , x 2 , x 3 ) , u r j ( x ) = u r ( x − y j ) for j = 0 , 1 , … , m − 1 . Setting r m = ( 2 m − 1 ) ( r + 1 ) , it is clear that supp u r j ⊂ B r m ( 0 ) . Obviously { u r j + } j = 0 m − 1 are linearly independent. Indeed, if w + = ∑ j = 0 m − 1 c j u r j + = 0 , denoting w = ∑ j = 0 m − 1 c j u r j , one has w = w − + w + and

− ∥ w − ∥ 2 = a ( w ) = ∑ j c j 2 a ( u r j + ) = a ( u r ) ∑ j c j 2 ,

which implies c j = 0 , j = 0 , 1 , … , m − 1 . Now set

E m = E − ⊕ span { u r j : j = 0 , … , m − 1 } = E − ⊕ span { u r j + : j = 0 , … , m − 1 } .

By virtue of Lemma 3.9, let t ε r j > 0 be such that t ε r j u r j + ∈ N ε . Observe that

(4.3) lim ε → 0 lim r → ∞ t ε r j = lim ε → 0 t ε j = 1 ;

(4.4) lim ε → 0 lim r → ∞ h ε ( t ε r j u r j + ) = lim ε → 0 h ε ( t ε j u r j + ) = h 0 ( u + ) = u − ;

(4.5) lim ε → 0 lim r → ∞ ∥ h ε ( t ε r j u r j + ) − t ε r j u r j − ∥ = lim ε → 0 ∥ h ε ( t ε j u + ) − t ε j u − ∥ = 0 .

It is not difficult to check the following:

max w ∈ E m Φ ε ( w ) = Φ ε ∑ j = 0 m − 1 t ε j u r j + + h ε ( t ε j u r j + ) = Φ ε ∑ j = 0 m − 1 t ε j u r j + + t ε j u r j − + o ( 1 r ) = Φ ε ∑ j = 0 m − 1 t ε j u r j + o ( 1 r ) = ∑ j = 0 m − 1 Φ ε ( t ε j u r j ) + o ( 1 r ) = ∑ j = 0 m − 1 Φ ε ( t ε j u r j + + t ε j u r j − ) + o ( 1 r ) = ∑ j = 0 m − 1 Φ ε ( t ε j u r j + + h ε ( t ε j u r j + ) ) + o ( 1 r ) = ∑ j = 0 m − 1 Φ 0 ( t 0 j u r j + + h 0 ( t 0 j u r j + ) ) + o ( 1 r ε ) = ∑ j = 0 m − 1 Φ 0 ( u ) + o ( 1 r ε ) = m γ λ ¯ , κ + o ( 1 r ε ) ,

where o ( 1 r ) means arbitrary small as r → ∞ , and o ( 1 r ε ) means arbitrary small as r is sufficiently large and ε is sufficiently small. Now, by assumptions and Lemma 3.1, for any 0 < δ < γ ∞ − m γ λ ¯ , κ , one may choose r > 0 large and then ε m > 0 small such that, for all ε ≤ ε m , max w ∈ E m Φ ε ( w ) ≤ γ ∞ − δ . Now by Theorem 2.1 one obtains the multiplicity conclusion.

Let ℒ ε denote the set of all least energy solutions of Φ ˜ ε . Let ε j → 0 , u j ∈ ℒ j , where ℒ j = ℒ ε j . Then { u j } is bounded. A standard concentration argument (see [27]) shows that there exist a sequence { x j } ⊂ R 3 and constants R > 0 , δ > 0 such that

liminf j → ∞ ∫ B ( x j , R ) ∣ u j ∣ 2 d x ≥ δ .

Set

v j = u j ( x + x j )

and denoted by λ ^ j ( x ) = λ ( ε j ( x + x j ) ) , W ^ j ( x ) = W ( ε j ( x + x j ) ) , one easily checks that v j solves

H ω v j − λ ^ j ( x ) V ε j , v j β v j = W ^ j ( x ) ⋅ ∣ v j ∣ p − 2 v j ,

with energy

S ( v j ) ≔ 1 2 ( ∥ v j + ∥ 2 − ∥ v j − ∥ 2 ) − Γ λ ^ j ( x ) ( v j ) − 1 p ∫ R 3 W ^ j ( x ) ∣ v j ∣ p d x = Φ ˜ j ( v j ) = Φ j ( v j ) = Γ λ ^ j ( x ) ( v j ) + 1 2 − 1 p ∫ R 3 W ^ j ( x ) ∣ v j ∣ p d x = c ε j .

Additionally, v j ⇀ v in E and v j → v in L loc q for q ∈ [ 1 , 3 ) . We now turn to prove that { ε j x j } is bounded. Arguing indirectly we assume ε j ∣ x j ∣ → ∞ and obtain a contradiction. Without loss of generality assume λ ( ε j x j ) → λ ∞ , W ( ε j x j ) → W ∞ . By the boundedness of ∇ λ and ∇ W , one sees that λ ^ j ( x ) → λ ∞ , W ^ j ( x ) → W ∞ uniformly on bounded sets of x . Since for any ψ ∈ C c ∞ ,

0 = lim j → ∞ ∫ R 3 ( H ω v j − λ ^ j ( x ) V ε j , v j β v j − W ^ j ∣ v j ∣ p − 2 v j ) ψ ¯ d x = ∫ R 3 ( H ω v − λ ∞ V v β v − W ∞ ∣ v ∣ p − 2 v ) ψ ¯ d x ,

hence v solves

i α ⋅ ∇ v − a β v + ω v − λ ∞ V v β v = W ∞ ∣ v ∣ p − 2 v .

Therefore,

S ∞ ( v ) ≔ 1 2 ( ∥ v + ∥ 2 − ∥ v − ∥ 2 ) − Γ λ ∞ ( v ) − W ∞ ∫ R 3 ∣ v ∣ p d x ≥ γ ˜ ∞ .

It follows from λ ¯ > λ ∞ , κ > W ∞ , one has γ ˜ λ ¯ , κ < γ ˜ ∞ . Moreover, by Fatou’s lemma, we have

γ ˜ λ ¯ , κ < γ ˜ ∞ ⩽ S ∞ ( v ) ⩽ lim j → ∞ c ε j = γ ˜ λ ¯ , κ ,

a contradiction. Thus, { ε j x j } is bounded. And hence, we can assume y j = ε j x j → y 0 . Then repeating the proof of [13] gives the concentration and the exponential decay.

Finally, by Lemma 3.14 we see that the solutions are in ⋂ q ≥ 2 W 1 , q .

4.2 Proof of Corollary 1.1

By Theorem 1.1 and Remark 1.1, we obtain the ground solution of Dirac-Klein-Gordon systems which converges to a ground state solution of the autonomuous Dirac-Klein-Gordon systems equation (1.4) for every c > 0 . Let { c n } , { ω n } be two real sequences such that

0 < c n , − ω n → + ∞ , 0 < − ω n < m c n 2 , m c n 2 + ω n → ν m ,

as n → ∞ . Let { ( ψ n , ϕ n ) } (where ψ n ≔ ( u n , v n ) ∈ C 4 ) denote a sequence of solutions for system equation (1.4) with frequency ω n at speed of light c n , then according to [20], there exists a mass m 0 , λ 0 , such that for m ≤ m 0 , λ ≤ λ 0 , up to a subsequence,

u n → 0 , v n → v , in H 1 ( R 3 , C 2 ) ,

ϕ n → 0 , in H 1 ( R 3 , R ) ,

as n → ∞ , where v : R 3 → C 2 is a solution for the following coupled system of nonlinear Schrödinger-type equations

− Δ v 1 + 2 ν v 1 = 2 m κ ∣ v ∣ p − 2 v 1 , − Δ v 2 + 2 ν v 2 = 2 m κ ∣ v ∣ p − 2 v 2 .

4.3 Proof of main results: the critical case

Part 1. Multiplicity

Without loss of generality, we may assume that 0 ∈ W 1 ∩ W 2 ∩ Γ , hence, λ ¯ = λ ( 0 ) , κ 1 = W 1 ( 0 ) , κ 2 = W 2 ( 0 ) . Solutions of (2.7) are critical points of the functional Φ ε ( u ) = Φ ε λ ¯ , κ → ( u ) . We will utilize Theorem 2.1. Obviously, Φ ε is even, and in virtue of Remark 2.1 the conditions ( Φ 1 ) and ( Φ 2 ) are satisfied. It remains to verify ( Φ 3 ) .

Let u be a critical point Φ 0 ≔ ϕ λ ¯ , κ → , with Φ 0 ( u ) = γ λ ¯ , κ → . Define u r , u r j , j = 0 , … , m − 1 , and set E m as before.

By virtue of Lemma 3.9, let t ε r j > 0 be such that t ε r j u r j + ∈ N ε . Observe that (4.3), (4.4), and (4.5) keep true. One then checks easily the following

where o ( 1 r ) means arbitrary small as r → ∞ , and o ( 1 r ε ) means arbitrary small as r is sufficiently large and ε is sufficiently small. Now, by assumptions and Lemma 3.1, for any 0 < δ < γ ∞ → − m γ λ ¯ , κ → , one may choose r > 0 large and then ε m > 0 small such that, for all ε ≤ ε m , max w ∈ E m Φ ε ( w ) ≤ γ ∞ → − δ . Now by Theorem 2.1, one obtains the multiplicity conclusion.

Part 2. Concentration

liminf j → ∞ ∫ B ( x j , R ) ∣ u j ∣ 2 d x ≥ δ .

Set

v j = u j ( x + x j )

and denoted by λ ^ j ( x ) = λ ( ε j ( x + x j ) ) , W ^ j 1 ( x ) = W 1 ( ε j ( x + x j ) ) , W ^ j 2 ( x ) = W 2 ( ε j ( x + x j ) ) one easily checks that v j solves

(4.6) H ω v j − λ ^ j ( x ) V ε j , v j β v j = W ^ j 1 ( x ) ∣ v j ∣ p − 2 v j + W ^ j 2 ( x ) ∣ v j ∣ v j ,

with energy

S ( v j ) ≔ 1 2 ( ∥ v j + ∥ 2 − ∥ v j − ∥ 2 ) − Γ λ ^ j ( x ) ( v j ) − ∫ R 3 W ^ j ( x ) ∣ v j ∣ p d x + ∫ R 3 W ^ j 2 ( x ) ∣ v j ∣ 3 d x = Φ ˜ j ( v j ) = Φ j ( v j ) = c ε j .

Additionally, v j ⇀ v in E and v j → v in L loc q for q ∈ [ 1 , 3 ) . We now turn to prove that { ε j x j } is bounded. Arguing indirectly we assume ε j ∣ x j ∣ → ∞ and obtain a contradiction. Without loss of generality assume λ ( ε j x j ) → λ ∞ , W 1 ( ε j x j ) → W 1 ∞ , W ^ j 2 ( ε j x j ) → W 2 ∞ . By the boundedness of ∇ λ and ∇ W , one sees that λ ^ j ( x ) → λ ∞ , W ^ j 1 ( x ) → W 1 ∞ , W ^ j 2 ( x ) → W 2 ∞ uniformly on bounded sets of x . Since for any ψ ∈ C c ∞ ,

0 = lim j → ∞ ∫ R 3 ( H ω v j − λ ^ j ( x ) V ε j , v j β v j − W ^ j 1 ( x ) ∣ v j ∣ p − 2 v j + W ^ j 2 ( x ) ∣ v j ∣ v j ) ψ ¯ d x = ∫ R 3 ( H ω v − λ ∞ V v β v − W 1 ∞ ∣ v ∣ p − 2 v − W 2 ∞ ∣ v ∣ v ) ψ ¯ d x ,

hence v solves

i α ⋅ ∇ v − a β v + ω v − λ ∞ V v β v = W 1 ∞ ∣ v ∣ p − 2 v + W 2 ∞ ∣ v ∣ v .

Therefore,

S ∞ ( v ) ≔ 1 2 ( ∥ v + ∥ 2 − ∥ v − ∥ 2 ) − Γ λ ∞ ( v ) − W 1 ∞ ∫ R 3 ∣ v ∣ p d x − W 2 ∞ ∫ R 3 ∣ v ∣ 3 d x ≥ γ ˜ ∞ .

It follows from λ ¯ > λ ∞ , κ > W ∞ , one has γ ˜ λ ¯ , κ < γ ˜ ∞ . Moreover, by Fatou’s lemma, we have

γ ˜ λ ¯ , κ < γ ˜ ∞ ≤ S ∞ ( v ) ≤ lim j → ∞ c ε j = γ ˜ λ ¯ , κ ,

a contradiction. Thus, { ε j x j } is bounded. And hence, we can assume y j = ε j x j → y 0 . Then v solves

(4.7) i α ⋅ ∇ v − a β v + ω v − λ ∞ V v β v = W 1 ( y 0 ) ∣ v ∣ p − 2 v + W 2 ( y 0 ) ∣ v ∣ v .

By Lemma 3.10, it is easy to check that lim ε → 0 dist ( ε y ε , W 1 ∩ W 2 ∩ Γ ) = 0 .

In order to prove v j → v in E , recall that as the argument shows

lim j → ∞ ∫ R 3 W ^ j 2 ( x ) ∣ v j ∣ 3 d x = ∫ R 3 W ( y 0 ) ∣ v ∣ 3 d x .

By the decay of v , using the Brezis-Lieb lemma, one obtains ∣ v j − v ∣ 3 → 0 . Hence, using the interpolation inequality and the boundedness of v j in E yields v j → v in L t ( R 3 , C 4 ) for t ∈ ( 2 , 3 ] . Denote z j = v j − v . The scalar product of (4.6) with z j + yields

( v j + , z j + ) = o ( 1 ) .

Similarly, using the decay of v together with the fact that z j ± → 0 in L loc q for q ∈ [ 1 , 3 ) , it follows from (4.7) that

( v + , z j + ) = o ( 1 ) .

Thus,

∥ z j + ∥ = o ( 1 ) ,

and the same arguments show

∥ z j − ∥ = o ( 1 ) ,

we then obtain v j → v in E , and the arguments in [13] show that v j → v in H 1 .

Part 3. Exponential decay

For the later use denote D = i α ⋅ ∇ and for u ∈ ℒ ε rewrite (2.7) as

D u = a β u − ω u + λ ε ( x ) V ε , u β u + W 1 ε ( x ) ∣ u ∣ p − 2 u + W 2 ε ( x ) ∣ u ∣ u .

Acting the operator D on the two sides and noting that D 2 = − Δ , we obtain

(4.8) Δ u = ( a + λ ε ( x ) V ε , u ) 2 u − ( ω − W 1 ε ∣ u ∣ p − 2 − W 2 ε ∣ u ∣ ) 2 u − D ( λ ε ( x ) V ε , u ) β u − D ( W 1 ε ∣ u ∣ p − 2 + W 2 ε ∣ u ∣ ) u .

Now define

sgn u = u ∣ u ∣ if u ≠ 0 , 0 if u = 0 .

By Kato’s inequality [19] there holds

Δ ∣ u ∣ ⩾ ℜ [ Δ u ( sgn u ) ] .

Note that

ℜ [ D ( W 1 ε ∣ u ∣ p − 2 + W 2 ε ∣ u ∣ ) u ( sgn u ) ] = 0 .

Then, we obtain

(4.9) Δ ∣ u ∣ ⩾ ( a + λ ε ( x ) V ε , u ) 2 ∣ u ∣ − ( ω − W 1 ε ∣ u ∣ p − 2 − W 2 ε ∣ u ∣ ) 2 ∣ u ∣ − ∣ D ( λ ε ( x ) V ε , u ) ∣ ⋅ ∣ u ∣ .

It follows from Hölder inequality and u ∈ L ∞ ( R 3 , C 4 ) that

∣ D ( λ ε ( x ) V ε , u ) ∣ ≤ C .

So, due to (4.9), there exists a constant M > 0 such that

Δ ∣ u ∣ ≥ − M ∣ u ∣ .

It then follows from the sub-solution estimate [25,31] that

∣ u ( x ) ∣ ≤ C 0 ∫ B 1 ( x ) ∣ u ( y ) ∣ d y ,

where C 0 is independent of x and ε .

To obtain the uniformly decay estimate for the semi-classical states, we first need the following result:

Lemma 4.1

Let v ε and V v ε be given in the proof of Part 2. Then ∣ v ε ( x ) ∣ and ∣ V v ε ( x ) ∣ vanish at infinity uniformly in ε > 0 small.

Proof

Similar to [13, 16], we can easily prove this lemma. We omit it here.□

And at this point, applying the maximum principle (see [29]), we easily have

Lemma 4.2

Let v ε ∈ E be given in the proof of Part 2, then v ε exponentially decays at infinity uniformly in ε > 0 small. More specifically, there exist C , c > 0 independent of ε such that

∣ v ε ( x ) ∣ ⩽ C e − c ∣ x ∣ .

Consequently, we infer that

∣ u ε ( x ) ∣ ⩽ C e − c ∣ x − x ε ∣ .

With the above arguments, we can easily prove Theorem 1.2.

Acknowledgement

The authors would like to thank the anonymous reviewer for her/his careful reading and for the valuable remarks that helped to improve the presentation of the paper.

Funding information: This work was supported by the National Natural Science Foundation of China (NSFC11871242).
Conflict of interest: The authors state no conflict of interest.

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Received: 2021-08-16

Revised: 2022-04-25

Accepted: 2022-05-09

Published Online: 2022-07-02

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

Multiplicity and concentration of semi-classical solutions to nonlinear Dirac-Klein-Gordon systems

Abstract

1 Introduction and main results

Theorem 1.1

Remark 1.1

Corollary 1.1

Remark 1.2

Theorem 1.2

Remark 1.3

Remark 1.4

2 The variational framework

Lemma 2.1

Lemma 2.2

Proof

Lemma 2.3

Proof

Corollary 2.1

Lemma 2.4

Theorem 2.1

Remark 2.1

3 Cut-off arguments

3.1 The limit equation: subcritical case

Lemma 3.1

Proof

3.2 The limit equation: critical case

Lemma 3.2

Proof

Lemma 3.3

Proof

Lemma 3.4

Proof

Lemma 3.5

Proof

3.3 The cut-off functionals

Lemma 3.6

Proof

Lemma 3.7

Proof

Lemma 3.8

Lemma 3.9

Lemma 3.10

Proof

Lemma 3.11

Proof

Lemma 3.12

Proof

Lemma 3.13

Proof

Lemma 3.14

4 Proof of main results

4.1 Proof of main results: the subcritical case

4.2 Proof of Corollary 1.1

4.3 Proof of main results: the critical case

Lemma 4.1

Proof

Lemma 4.2

Acknowledgement

References

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