Solitary waves for nonlinear Klein–Gordon equations coupled with Born–Infeld theory

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Solitary waves for nonlinear Klein–Gordon equations coupled with Born–Infeld theory

Yong Yu

Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA Received 21 February 2009; received in revised form 28 September 2009; accepted 5 October 2009

Available online 10 November 2009

Abstract

We consider the nonlinear Klein–Gordon equations coupled with the Born–Infeld theory under the electrostatic solitary wave ansatz. The existence of the least-action solitary waves is proved in both bounded smooth domain case andR³case. In particular, for bounded smooth domain case, we study the asymptotic behaviors and profiles of the positive least-action solitary waves with respect to the frequency parameterω. We show that whenκandωare suitably large, the least-action solitary waves admit only one local maximum point. Whenω→ ∞, the point-condensation phenomenon occurs if we consider the normalized least-action solitary waves.

©2009 Elsevier Masson SAS. All rights reserved.

1. Introduction

1.1. Background

The Born–Infeld geometric theory of electromagnetism is a nonlinear generalization of the classical Maxwell theory. It was introduced to overcome the infinite energy problem associated with a point-charge source in the original Maxwell theory. Based on the action principle of special relativity, Born proposed thefirstBorn–Infeld theory (cf. [2]

and [3]). It is defined by the action density LBI,1=b²

1−

1+F_μνF^μν 2b²

−det(gμν) (1.1)

wheregμν is the metric tensor of a(3+1)-dimensional Minkowskian space–time of the signature(+ − − −), and F_μν =∂_μA_ν −∂_νA_μ is the field strength curvature induced from a gauge potential (connection 1-form)A_μ. Later, by considering the invariance principle, Born and Infeld reconsidered (1.1) and introduced thesecondBorn–Infeld theory in [8] and [9], which is defined by the action density

LBI,2=b²

−det(gμν)−

−det

gμν+F_μν

b . (1.2)

E-mail address:yuyong@cims.nyu.edu.

0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2009.11.001

It is clear that when the Born–Infeld free parameter b→ ∞, both (1.1) and (1.2) reduce to the classical Maxwell theory.

In recent years, the Born–Infeld nonlinear electromagnetism has regained its importance due to its relevance in the theory of superstrings and membranes (cf. [29]). It has received much attention from both theoretic physicists and mathematicians (cf. [32,27,1,18,34,24]). Mathematically, motivated from Gibbons’ work (cf. [18]), in [34], Yang proposed an extended Born–Infeld equation, which can be used to unify the minimal surface equations and maximal hypersurface equations. Meanwhile, the author studied the existence of magnetostatic minimum-energy solutions in the Born–Infeld–Higgs model. Later, in Lin and Yang’s work (cf. [24]), the authors studied the gauged harmonic maps by extending the scalar Higgs fields in [34] to maps from a 2-surface into the standard 2-sphere. The coexistence of vortices and antivortices were obtained by studying the first-order system of self-dual and anti-self-dual equations.

The existence of cosmic strings induced by these vortices were also established.

1.2. Electrostatic solitary wave ansatz

As we know, the gauge potentialA_μcan be coupled to a complex order parameterψthrough the minimal coupling rule. That is the formal substitution

∂

∂t → ∂

∂t −iA0,

∇ → ∇ −iA

where A=(A1, A2, A3) is a magnetic vector potential, and A0 is an electric potential. Therefore, in a flat Minkowskian space–time with metric(g_μν)=diag[1,−1,−1,−1], we can define the Klein–Gordon–Maxwell La- grangian densityLKGMas

LKGM=1 2

∂ψ

∂t −iψ A0

²− |∇ψ−iAψ|²−m²|ψ|² +1

p|ψ|^p (1.3)

wherem is the mass of a particle, andp∈(2,6) is a constant describing the nonlinearity in (1.3). The nonlinear Born–Infeld–Klein–Gordon equations (NBIKG for short) are the Euler–Lagrange equations of the total action

S=

LBI,1+LKGM. (1.4)

Under the electrostatic solitary wave ansatz

ψ (x, t )=u(x)e^iωt, A0= −φ(x), A=0

whereuandφare real-valued functions defined on a subset U ofR³, andωis a positive frequency parameter, the total action in (1.4) takes the form

F_U(u, φ):=

U

1

2|∇u|²+1 2

m²−ω² u²−1

p|u|^pdx−E_u,U(φ) (1.5)

with

E_u,U(φ):=

U

b²

1−

1− 1 b²|∇φ|²

+ωu²φ+1

2φ²u². (1.6)

The critical point (φ,u) ofF_U satisfies the Euler–Lagrange equations associated with (1.5). By standard calculations, we get

⎧⎪

⎪⎨

⎪⎪

⎩

∇ · ∇φ

1−_b¹2|∇φ|²=u²(ω+φ), u=

m²−(ω+φ)²

u− |u|^p−2u.

(1.7)

In this article,Uis assumed to be a bounded smooth domain inR³or the wholeR³space. WhenU=R³, we omit the subscriptionUfrom (1.5) and (1.6). That is, we useF andEuto denoteF_R3 andE_u,R3, respectively. As a convention in this paper, in an integral expression, if its integration domain isR³, we will omitR³from the integral expression.

1.3. Main results

The existence of solitary waves has been well studied in different systems (cf. [4–7,10–13,16,22,25]). Particularly in [13] and [25], the authors considered the system of nonlinear Klein–Gordon equation coupled with the Born–Infeld type equations. Here, the Born–Infeld type equations refer to the second order expansion of the original Born–Infeld equations when the Born–Infeld parameterb→ ∞. In this article, we study the original Born–Infeld equations.

Firstly, for any fixedu∈L²(U ), we study the variational problem

φmin∈ME_u,U(φ). (1.8)

The configuration space M:=D(U )∩

φ∇φL^∞(U )b

(1.9) naturally arises from the physical constraint (quantity under the square root in (1.6) should be nonnegative) and the finite energy condition (Eu,U(φ) <∞) of (1.6). In (1.9),D(U )denotes the completion ofC₀^∞(U;R)with respect to the norm

φD(U ):= ∇φL²(U )+ ∇φL⁴(U ).

It is embedded intoL^∞(U )continuously (cf. Proposition 8 in [17]). Therefore,Mis a topological space by equipping with the uniform norm topology whenU is a bounded smooth domain or the locally uniform norm topology when U=R³. The existence of global minimizer for the variational problem (1.8) will be studied in Section 2.1 by an application of the direct method in the Calculus of Variations. The minimizer is unique because of the convexity of the functionalE_u,U. Therefore, we can define a nonlinear operator

Φ:L²(U )→M, (1.10)

which sends one L²(U ) functionu to the unique minimizer of the variational problem (1.8). In Section 2.3, the operatorΦis proved to be a continuous map betweenL²(U )andMif we equipL²(U )with its strong topology.

Plug the minimizerΦ(u)into (1.5), the functional J[u] :=FU

u, Φ(u)

, (1.11)

which is defined onH¹(U )or a subspace ofH¹(U ), is strongly indefinite. In Proposition 3.1, the functionalJ will be proved to beC¹differentiable in the sense of Fréchet. We emphasize here that, with the loss of theC¹regularity onΦ, our method relies only on the continuity of the operatorΦ. In Section 3.2, we will assumeU=R³. By theZ2

Mountain Pass Theorem, we will prove the existence of infinite many critical points ofJ with radial symmetry when κ=m

ω (1.12)

is suitably large (see Theorem 3.3). Furthermore, we will show that among all nonzero critical points ofJwith radial symmetry, there exists one that attains the least-J-action (see Theorem 3.8). In Section 3.3,Uis a bounded smooth domain. We will study the positive critical points of the functionalJ with boundary values set to be 0. Therefore, we work on the functional

J₊[u] :=

U

1

2|∇u|²+1 2

m²−ω² u²− 1

pu^p₊dx−E_u,U Φ(u)

. (1.13)

J₊ is defined onH₀¹(U ). Based on the well-known Mountain Pass Lemma due to Ambrosetti and Rabinowitz, we will show in Theorem 3.18 and Theorem 3.19 that ifκ is suitably large, then among all nonzero critical points ofJ₊, there exists one that attains the least-J-action. We call it the positive least-J-action critical point ofJ. Compared to the magnetostatic minimum-energy solutions for the Born–Infeld–Higgs model or the gauged harmonic map model in [34] and [24], our results in Theorems 3.3, 3.8, 3.18 and 3.19 verify the existence of the least-action solutions in the context of electrostatic fields. Moreover, in Theorem 3.18, we generalize the existence of the Mountain Pass solution in [26] and [33] to a coupled system combining a semilinear elliptic equation with an electric potential governed by the Born–Infeld theory.

Denote u_κ,ω by one positive least-J-action critical point of the functional J₊. With respect to the parameters κ and ω, in Section 4, we focus on the asymptotic behaviors and profiles of the functions in relation to u_κ,ω andΦ_ω(u_κ,ω). We adopt a singular perturbation method. In fact, in Section 4.3, we will prove the point-condensation phenomenon for the normalized functionv_κ,ω=ω^2−2pu_κ,ω. More precisely, it will be proved that whenκis a suitably large constant, in the limit ofω→ ∞,vκ,ωadmits only one local maximum pointPκ,ω. In particular,

v_κ,ω(P_κ,ω)

κ²−1 ¹

p−2 (1.14)

is uniformly bounded from below by a constant depending only onκ. Meanwhile, the normalized functionvκ,ω→0 inC_loc^1,α(U\P_κ,ω).

2. Minimizer ofEu,U

2.1. Existence of the minimizerΦ(u)

DenoteD^1,2(U )by the completion ofC₀^∞(U;R)with respect to the norm φD^1,2(U ):= ∇φL²(U ).

D(U )is continuously embedded into bothD^1,2(U )andL^∞(U ). Moreover,D^1,2(U )is continuously embedded into L⁶(U )by Sobolev inequality.

In the following, we begin to study the variational problem (1.8) in this section. First, let us state a lemma.

Lemma 2.1.If{φ_n} ⊆M andφ_n φinD^1,2(U ), thenφ_n→φuniformly inU¯ ifUis a bounded smooth domain;

φ_n→φlocally uniformly inR³, ifU=R³.

Proof. Assume that{φ_nk}is any subsequence of{φ_n}. Since{φ_nk} ⊆Mandφ_nk φ weakly inD^1,2(U ),{∇φ_nk}is uniformly bounded inL²(U )andL⁴(U ). IfUis a bounded smooth domain, by Morrey’s inequality, we then have

φn_k

C^0,14(U )¯ Cφn_kD(U ),

which implies that {φ_nk}is equicontinuous on U. Apply Arzelá–Ascoli’s theorem, we can extract a subsequence,¯ which is denoted by{φn_kl}, such thatφn_kl ⇒φinU. Because the subsequence¯ {φn_k}is arbitrary, we haveφn⇒φ inU.¯ R³case is similar. We omit the proof. 2

In the following, we use the direct method in the Calculus of Variations (cf. Theorem 1.2 in [31]) to show that Theorem 2.2.For everyu∈L²(U ), there exists a uniqueΦ(u)∈Msuch that

E_u,U Φ(u)

= inf

φ∈ME_u,U(φ).

In fact, we need to verify that:

(1) Mis a weakly closed subset ofD^1,2(U );

(2) E_u,U is coercive and weakly sequentially lower semicontinuous onMwith respect toD^1,2(U ).

Lemma 2.3.Mis a weakly closed subset ofD^1,2(U ).

Proof. Note that,D^1,2(U )is reflexive andMis convex. We only need to prove the strong closedness ofMinD^1,2(U ).

Choose{φn} ⊆M,φn→φinD^1,2(U ). Up to a subsequence, we can assume∇φn→ ∇φalmost everywhere inU.

Then∇φL^∞(U )b. We also have

U

|∇φ_n− ∇φ|⁴4b²

U

|∇φ_n− ∇φ|²→0.

Hence,φ∈D(U ). Under the definition ofMin (1.9),φ∈M. 2 Pay attention to the fact that

F (x, φ, p)=b²

1−

1− 1 b²|p|²

+ωu²(x)φ+1

2u²(x)φ²−1

2ω²u²(x)

is convex inp. Then,F is a Caratheodory function. In the following, we apply Theorem 1.6 in [31] and show that Lemma 2.4.E_u,U(·)is coercive and weakly sequentially lower semicontinuous with respect toD^1,2(U ).

Proof. The coercivity ofE_u,U with respect toD^1,2(U )is a result of the fact that E_u,U(φ)

U

1

2|∇φ|²dx−1 2ω²

U

u²dx.

Let us now prove the weakly sequentially lower semicontinuity ofE_u,U. Assume that{φ_n} ⊆Mandφ_n φweakly inD^1,2(U ). From Lemma 2.1, we know thatφn⇒φlocally inU. That is,∀UU, we haveφn⇒φinU. Hence, φn→φinL¹(U). Since∇φn∇φweakly inL²(U ), we get∇φn∇φweakly inL¹(U). Apply Theorem 1.6 in [31],

Eu,U(φ)lim inf

n→∞ Eu,U(φn). 2

Remark 2.5.Use the same method as the above, we can prove that if{φ_n} ⊆Mandφ_n φweakly inD^1,2(U ), then

U

b²

1−

1− 1 b²|∇φ|²

lim inf

n→∞

U

b²

1−

1− 1 b²|∇φ_n|²

.

Now we complete this section by the proof of Theorem 2.2.

Proof of Theorem 2.2. Lemma 2.3 and Lemma 2.4 together with Theorem 1.2 in [31] imply that there exists a minimizer ofE_u,U onM. The convexity of the functionalE_u,U ensures the uniqueness of the minimizer. 2 2.2. Some properties of the operatorΦ

From Theorem 2.2, we can construct the operatorΦ, which is defined as in (1.10). Since for a fixedu∈L²(U ), Φ(u)is the unique minimizer of the variational problem (1.8), then

Proposition 2.6.∀u∈L²(U )fixed,Φ=Φ(u), we have

U

|∇Φ|²

1−_b¹2|∇Φ|²

U

−u²(ω+Φ)Φ. (2.1)

Proof. ∀λ∈(0,1),λΦ∈M. ConsiderE_u,U(λΦ)as a function ofλ, then d

dλE_u,U(λΦ)=

U

λ |∇Φ|² 1−_b¹2|∇Φ|²λ²

+ωu²Φ+λu²Φ². (2.2)

SinceΦattains the minimum ofE_u,U inM, we have lim inf

λ→1⁻

d

dλE_u,U(λΦ)0. (2.3)

Apply Fatou’s Lemma, (2.2) and (2.3) may imply (2.1). 2

SinceD(U )is embedded intoL^∞(U )continuously, then(ω+Φ)Φ∈L^∞(U ). By the assumption thatu∈L²(U ), we can conclude from Proposition 2.6 that the minimizerΦ=Φ(u)of the variational problem (1.8) satisfies

U

|∇Φ|²

1−_b¹2|∇Φ|²<+∞. Therefore, we can defineK⊆Mas

K:=

ψ∈M

U

|∇ψ|²

1−_b¹2|∇ψ|²

<+∞

. (2.4)

The minimizerΦ=Φ(u)of the variational problem (1.8) is also the minimizer of the problem to minimize the energy functionalE_u,U onK. Therefore, we may get the following proposition.

Proposition 2.7(Variational Inequality).∀u∈L²(U )fixed,Φ=Φ(u)is the minimizer ofE_u,UonM. Then,∀ψ∈K,

U

∇Φ

1−_b¹2|∇Φ|²· ∇(Φ−ψ )+(ω+Φ)(Φ−ψ )u²0. (2.5)

Proof. ∀ψ∈Kand 0< λ <1,λΦ+(1−λ)ψ∈K. We know thatΦ(u)attains the minimum ofEu,U onK, thus, lim inf

λ→1⁻

d dλE_u,U

λΦ+(1−λ)ψ 0.

Apply Lebesgue’s Dominated Convergence theorem. We complete the proof. 2

As an application of the variational inequality (2.5), we prove the a priori upper and lower bounds ofΦ(u)for a fixedu∈L²(U ).

Proposition 2.8.∀u∈L²(U ), it results inΦ(u)0. Moreover,Φ(u)(x)−ωifu(x)=0.

Proof. Apply the variational inequality (2.5) and setψ= −Φ⁻, we have

U

∇Φ

1−_b¹2|∇Φ|²· ∇

Φ+Φ⁻ +ω

Φ+Φ⁻ u²+

Φ+Φ⁻

Φu²0.

That is,

Φ0

∇Φ⁺

1−_b¹2|∇Φ⁺|²· ∇Φ⁺+ωΦ⁺u²+Φ⁺Φ⁺u²0.

So we get∇Φ⁺=0. Hence,Φ0. If we setψ= −ω+(ω+Φ)⁺, then we have

Φ−ω

|∇Φ|²

1−_b¹2|∇Φ|²+u²(Φ+ω)²0. (2.6)

Hence,Φ(u)(x)−ωwheneveru(x)=0. 2

In the rest of this section, we assume thatU=R³. Letf be a function defined onR³. With respect to fixedx₀∈R³ andg∈O(3), we can define translation and rotation onf as

T_x0f (x):=f (x+x₀), T_gf (x):=f (gx), ∀x∈R³. (2.7) From the definition ofK in (2.4), it is easy to prove thatK is translation–rotation invariant. Pay attention to the uniqueness result in Theorem 2.2. We prove the fact thatΦ is even and commutes with the group of rototranslations in Proposition 2.9 below.

Proposition 2.9.∀u∈L²(R³),∀g∈O(3),∀x₀∈R³, we have

Φ(T_x0u)=T_x0Φ(u), (2.8)

Φ(T_gu)=T_gΦ(u), (2.9)

Φ(u)=Φ(−u). (2.10)

Proof. We know that E_u

Φ(u)

=E_Tx

0u

T_x0Φ(u) .

SinceK is translation–rotation invariant, we haveT_x0Φ(u)∈K. Note that,Φ(T_x0u)attains the minimum ofE_Tx

0u

onM. Then, E_u

Φ(u) E_Tx

0u

Φ(T_x0u) . Sincex₀is arbitrary, we conclude that

E_u Φ(u)

=E_Tx

0u

Φ(T_x0u)

=E_Tx

0u

T_x0Φ(u) .

The uniqueness result in Theorem 2.2 immediately implies (2.8). The proofs for (2.9) and (2.10) are similar. We omit the discussions here. 2

As an application of (2.8) in Proposition 2.9 and the variational inequality (2.5) in Proposition 2.7, let us consider aW^2,2estimate on the minimizerΦ(u).

Lemma 2.10. If ∇Φ(u)∈ L^p1(R³), ∇u∈L²(R³)∩L^p2(R³) and u∈L²(R³)∩L^p3(R³) where 1p_i ∞ (i=1,2,3)satisfying _p¹

1 +_p¹₂ +_p¹₃ =1, then

i,j

∂_ijΦ(u)²6ω∇Φ(u)

L^p1(R³)∇uL^p2(R³)uL^p3(R³). (2.11) Proof. Assume that Φ =Φ(u), Φ1, :=Φ(x+e1) and u1, :=u(x+e1) where e1=(1,0,0). Since K is translation–rotation invariant,Φ1,∈K. By Proposition 2.7, we have

∇Φ

1−_b¹2|∇Φ|²· ∇(Φ−Φ_1,)+ω(Φ−Φ_1,)u²+(Φ−Φ_1,)Φu²0. (2.12) From (2.8) in Proposition 2.9, we know thatΦ1,=Φ(u(· +e1)). Apply Proposition 2.7 again, we have

∇Φ_1,

1−_b¹2|∇Φ1,|²· ∇(Φ_1,−Φ)+ω(Φ_1,−Φ)u²_1,+(Φ_1,−Φ)Φ_1,u²_1,0. (2.13) By adding the inequalities (2.12) and (2.13), we have

∇Φ_1,−Φ

²+

ω+Φ(u)Φ_1,−Φ

u²(x+e₁)−u²(x)

0.

Therefore,

∇Φ_1,−Φ

²+2u

ω+Φ(u)Φ_1,−Φ

u(x+e₁)−u(x)

+ ω+Φ(u)Φ_1,−Φ

u(x+e₁)−u(x)

u(x+e1)−u(x)

0. (2.14)

Since we know that if 1p∞, Φ_1,−Φ

L^p(R³)

∇ΦL^p(R³). In particular, ifp= ∞,

Φ1,−Φ

L^∞(R³)

b. (2.15)

Then, we know that

ω+Φ(u)Φ1,−Φ

u(x+e₁)−u(x)

2

bω+Φ(u)

L^∞(R³)∇u²_L2(R³). Therefore, from (2.14) and Proposition 2.8, we have

∇Φ1,−Φ

²2

ω+Φ(u)Φ1,−Φ

u(x+e1)−u(x)

u

+O() 2ω

Φ1,−Φ

u(x+e1)−u(x)

u

+O(). (2.16)

By applying Hölder’s inequality on (2.16) and letting→0, we get ∇∂₁Φ(u)²2ω∇ΦL^p1(R³)∇uL^p2(R³)uL^p3(R³)

where 1p_i∞(i=1,2,3)satisfying _p¹

1 +_p¹₂ +_p¹₃ =1. 2

Apply Lemma 2.10 by settingp₁= ∞,p₂=p₃=2. The following proposition holds:

Proposition 2.11.Ifu∈H¹(R³), then∇Φ(u)∈H¹(R³;R³), which satisfies

i,j

∂ijΦ(u)²6ωb∇uL²(R³)uL²(R³). (2.17)

2.3. Continuity of the operatorΦ

In this section, we study the continuity of the operator Φ in two cases. In Proposition 2.12, we prove that Φ:L²(U )→M is a continuous map if we equipL²(U )with the strongL²(U )topology. If we restrict the opera- torΦon

L:=L²(U )∩L¹²⁵(U )∩L³(U ) (2.18)

and equip L with the strong L¹²⁵(U )∩L³(U ) topology, then in Proposition 2.13, we show that Φ :L→M is a continuous map. Here,u_n→uunder the strongL¹²⁵(U )∩L³(U )topology means thatu_n→uin bothL¹²⁵(U )and L³(U ) strongly. As before,M is a topological space equipped with the uniform norm topology ifU is a bounded smooth domain or with the locally uniform norm topology ifU=R³.

Proposition 2.12.Φ:L²(U )→Mis a continuous map.

Proof. Assume thatu_n→ustrongly inL²(U ). In the following, we show that∀{u_nk} ⊆ {u_n}, there exists a subse- quence{u_nkl}, such that whenUis a bounded smooth domain,Φ(u_nkl)⇒Φ(u)inU¯; whenU=R³,Φ(u_nkl)⇒Φ(u) locally inR³.

Note that,{un_k}is uniformly bounded inL²(U ). By Propositions 2.6 and 2.8,{Φ(un_k)}is uniformly bounded in D(U ). Hence, it is also uniformly bounded inL^∞(U ). Then, there exist a subsequence{u_n

kl}andf ∈M, such that Φ(u_nkl) f inD^1,2(U ). By Lemma 2.1,Φ(u_nkl)⇒f inU¯ whenUis a bounded smooth domain orΦ(u_nkl)⇒f locally inR³whenU=R³. For the rest of the proof, we only need to show thatf =Φ(u).

BecauseΦ(u_n

kl)is the minimizer ofE_unk

l,U onMandu_n

kl →uinL²(U ), we have E_unk

l,U

Φ(u_n

kl) E_unk

l,U

Φ(u)

→E_u,U Φ(u)

. Note that,{Φ(u_n

kl)}is uniformly bounded inL^∞(U )and converges tof pointwisely. Whenu_n

kl→uinL²(U ), we have

U

Φ(un_kl)u²_n

kl→

U

f u²,

U

Φ(un_kl)²u²_n

kl→

U

f²u². (2.19)

From Remark 2.5 and (2.19), we get Eu,U(f )lim inf

l→∞ Eu_nk

l,U

Φ(un_kl) lim

l→∞Eu_nk

l,U

Φ(u)

=Eu,U

Φ(u) . By the uniqueness result in Theorem 2.2,f=Φ(u). 2

Proposition 2.13.Φ:L→Mis a continuous map.Lis defined in(2.18).

Proof. Assume thatu_n→uinL. WhenU is a bounded smooth domain, the result has already been included in Proposition 2.12. Now, we assumeU=R³. Notice Proposition 2.8 and apply Hölder’s inequality on the right-hand side of (2.1). We have

∇Φ(u_n)²+ 1

2b²∇Φ(u_n)⁴ωu_n²

L¹²⁵(R³)

Φ(u_n)

L⁶(R³). (2.20)

Sinceu_n→uinL, we have{u_n}uniformly bounded in L¹²⁵(R³). From (2.20),{Φ(u_n)}is uniformly bounded in D(R³). Hence, it is also uniformly bounded inL^∞(R³)andL⁶(R³). Then, there exist a subsequence, still denoted by {Φ(u_n)}, andf ∈M, such thatΦ(u_n) f inD^1,2(R³). Apply Lemma 2.1,Φ(u_n)⇒f locally inR³. For the rest of the proof, we only need to show thatf =Φ(u).

Note that,

u²_nΦ(u)−u²Φ(u)

u²_n−u²

L⁶⁵

Φ(u)

L⁶C∇Φ(u)

L²un−u

L¹²⁵ →0, (2.21)

u²_nΦ(u)²−u²Φ(u)²

u²_n−u²

L³²

Φ(u)²

L⁶C∇Φ(u)²

L²un−uL³→0. (2.22) Hence, we have

nlim→∞E_un Φ(u)

=E_u Φ(u)

. SinceE_un(Φ(u_n))E_un(Φ(u)), then

lim sup

n

E_un Φ(u_n)

E_u Φ(u)

. (2.23)

In another way, note that,{Φ(u_n)}is uniformly bounded inL⁶(R³),L^∞(R³)and converges tof pointwisely. When u_n→uinL, we can apply the Lebesgue’s Dominated Convergence theorem and similar arguments as in (2.21) and (2.22) to show that

u²_nΦ(u_n)→

u²f,

u²_nΦ(u_n)²→

u²f². (2.24)

Because of Remark 2.5 and (2.24), we have lim inf

n Eu_n

Φ(un)

Eu(f ). (2.25)

Obviously, (2.23) and (2.25) imply thatE_u(f )=E_u(Φ(u)). Therefore, we havef =Φ(u)by the uniqueness result in Theorem 2.2. 2

Remark 2.14.Ifun→uinL²(U )orL, then

nlim→∞E_un,U Φ(u_n)

=E_u,U Φ(u)

. (2.26)

3. Existence of solitary wave solutions

In this section, we prove the existence of critical points ofJ andJ₊. Firstly, let us study theC¹differentiability of J andJ₊in the sense of Fréchet.

3.1. Differentiability ofJ andJ₊

In this section, the functionalJ is defined onH, which is a subspace ofH¹(U ). Naturally,His endowed with the metric fromH¹(U ). The main purpose of this section is to prove

Proposition 3.1.J∈C¹(H;R)in the sense of Fréchet.

Proof. It is sufficient to prove

vlim_H1→0

J[u+v] −J[u] −DJ[u]v

/vH¹=0 where

DJ[u]v:=

U

∇u· ∇v+ m²−

ω+Φ(u)2

uv− |u|^p−2uv. (3.1)

We splitJ[u+v] −J[u] −DJ[u]vinto three parts. That is, J[u+v] −J[u] −DJ[u]v=A+B+C

where A=

U

1

2∇(u+v)²−1

2|∇u|²− ∇u· ∇v, B= −1

p

U

|u+v|^p− |u|^p−p|u|^p−2uv,

C=Eu, U Φ(u)

−Eu+v,U

Φ(u+v) +

U

1 2

m²−ω² v²+

2ω+Φ(u)

Φ(u)uv.

By Sobolev embedding theorem, it is easy to prove A, B=o(vH¹). We thus focus on the proof of C=o(vH¹).

In the following, we will prove Co(vH¹)and Co(vH¹)successively. Then we complete the proof of Propo- sition 3.1.

We first prove Co(vH¹(R³)). SinceΦ(u+v)minimizesE_u+v,UonM, we have CD:=E_u,U

Φ(u)

−E_u+v,U Φ(u)

+

U

1 2

m²−ω² v²+

2ω+Φ(u)

Φ(u)uv.

Therefore, Co(v_H¹)by the fact that D=

U

1 2

m²−ω²

v²−ωΦ(u)v²−1

2Φ(u)²v²=o vH¹

.

Let us now prove Co(v_H¹). SinceΦ(u)minimizesE_u,U onM, CEu,U

Φ(u+v)

−Eu+v,U

Φ(u+v) +

U

2ω+Φ(u)

Φ(u)uv+1 2

m²−ω² v².

Set

Θ:=uv

Φ(u+v)−Φ(u)

2ω+Φ(u)+Φ(u+v) , we get

Co v_H¹

−

U

Θ. (3.2)

In the following, we prove

vlim_H1→0

1 vH¹

U

Θ

=0 (3.3)

in two cases, then from (3.2), we get Co(vH¹).

Case 1(Bounded Smooth Domain).Under definition ofΘ, we have by applying Hölder’s inequality that 1

v_H¹

U

Θ

2ω+Φ(u)+Φ(u+v)

L^∞(U )Φ(u+v)−Φ(u)

L^∞(U )uL²(U ).

From Propositions 2.6, 2.8 and the fact thatD(U )is embedded intoL^∞(U )continuously, we know that, asv→0 inH¹(U ),2ω+Φ(u)+Φ(u+v)L^∞(U )is uniformly bounded by a constant depending onb,ωanduL²(U ). By Proposition 2.12, we get (3.3) in the bounded smooth domain case.

Case 2(The whole Euclidean space R³). Similarly as in Case 1, whenv→0 in H¹(R³),Φ(u+v)_L∞(R³) is uniformly bounded by a constantC=C(b, ω,uL²(R³)). Sinceu∈L²(R³),∀ >0, there existsR >0 large enough, such thatuL²(B^c_R)< . Therefore, by Hölder’s inequality,

1 vH¹(R³)

B^c_R

Θ

CuL²(B^c_R)< C. (3.4)

FixRabove. Since we know from Proposition 2.12 that ifv→0 inL²(R³), thenΦ(u+v)⇒Φ(u)inB¯_R. We thus get

1 vH¹(R³)

BR

Θ

CΦ(u+v)−Φ(u)

L^∞(B¯_R)uL²(R³)→0, asvH¹(R³)→0. (3.5) By (3.4) and (3.5), we get (3.3) whenU=R³. 2

Remark 3.2.If we consider the functionalJ₊defined as in (1.13), then by using the same method as in the proof of Proposition 3.1, we getJ₊∈C¹(H;R)in the sense of Fréchet.

3.2. Solitary wave solutions –U=R³

The functionalJ in (1.11) is defined onH¹(R³). According to Proposition 3.1,J isC¹differentiable in the sense of Fréchet. In this section, we study the existence of critical points ofJ by theZ2 Mountain Pass Theorem. More precisely, we prove

Theorem 3.3. If (^p₂ −1)m²> ^p₂ω², then in H¹(R³), there exist infinitely many critical points of J with radial symmetry. Moreover, the critical points ofJ satisfy the equation

−u+ m²−

ω+Φ(u)2

u− |u|^p−2u=0. (3.6)

Before we prove Theorem 3.3, let us state two lemmas, which are simple applications of (2.9) and (2.10).

Lemma 3.4.∀u∈H¹(R³),J[u] =J[−u].

Lemma 3.5.∀u∈H¹(R³),∀g∈O(3), we haveJ[T_gu] =J[u].

Lemma 3.4 and Lemma 3.5 imply that the functionalJ is even andTg invariant. Therefore, by the principle of symmetric criticality (cf. [28]), we may restrictJon the radially symmetric subspaceH_r¹(R³)⊂H¹(R³). That is, Lemma 3.6.Ifu∈H_r¹(R³)is a critical point ofJ|_Hr¹_(R³₎, thenuis a critical point ofJ.

BecauseJ is invariant under translations, there is a lack of compactness onH¹(R³). For this reason, we restrict J to the subspaceH_r¹(R³), which is a natural constraint forJ in the sense of Lemma 3.6. Then, there is no lack of compactness. Indeed, we have

Lemma 3.7.If(^p₂−1)m²>^p₂ω², the functionalJ|H_r¹(R³)satisfies the Palais–Smale condition.

After some slight modifications, the proof of Lemma 3.7 is almost the same as the proof of Lemma 6 in [13]. We omit the discussion here. Now we begin to prove Theorem 3.3.

Proof of Theorem 3.3. Jis even. According to Lemma 3.7, we know thatJ|H_r¹satisfies the Palais–Smale condition.

Therefore, by Theorem 9.12 in [30], we only need to show thatJ|H_r¹ satisfies the following two geometric hypothesis:

(G₁) ∃ρ >0 andα >0 such thatJ[u]α,∀uwithu_Hr¹=ρ;

(G2) for every finite dimensional subspaceV ofH_r¹,∃R=R(V ) >0 such thatJ[u]0,∀u∈V withuH_r¹R. Since we know that∀u∈H¹(R³),

C(m, ω)u²_H1−1

pu^p_L^pJ[u]C(m)u²_H1−1 pu^p_L^p.

Sobolev inequality implies that uL^pCuH¹. This helps us to prove (G₁). In finite dimensional subspace V of H_r¹(R³), theL^p norm andH¹ norm are equivalent. Thus inV,uH¹ C(V )uL^p,∀u∈V. This helps us to complete the proof ofG2. 2

Theorem 3.3 implies the existence of infinitely many critical points of the functionalJ with the radial symmetry.

In Theorem 3.8 below, we prove the existence of least-J-action critical point among all nonzero critical points ofJ with the radial symmetry. Note thatJ is strongly indefinite. It is not bounded from above or below, we need to restrict our functionalJon the manifold

Σ:=

u∈H_r¹

R³ u≡0, uis a critical point ofJ .

Theorem 3.8.If(^p₂−1)m²>^p₂ω², then F := inf

u∈ΣJ[u] =min

u∈ΣJ[u]>0. (3.7)

Proof. First, let us proveF >0. Ifu∈Σ, then

|∇u|²+ m²−

ω+Φ(u)2

u²− |u|^p=0. (3.8)

Apply Sobolev embedding theorem,

|u|^p=

|∇u|²+ m²−

ω+Φ(u)2 u²C

|u|^{p 2}

p

(3.9) whereC >0 is a constant depending onmandω. Sinceu=0, we have

|u|^pC^pp−2 >0. (3.10)

That is,Σkeeps strictly away from 0. By (3.8) and the fact thatE_u(Φ(u))0, we have J[u] 1

2 −1 p

|∇u|²+ 1

2−1 p

m²−1

2ω² u²Cu²L^pC(p, ω, m) >0. (3.11) Hence,F >0.

If {un} ⊆Σ such that J[un] →F, then {J[un]} is bounded and from (3.11), {un} is uniformly bounded in H_r¹(R³). Therefore, we can extract a subsequence, denoted also by {un}, such that un u0 in H_r¹(R³). Since H_r¹(R³) →L^s(R³)compactly withs∈(2,6), we can further assume thatu_n→u₀inL^s(R³)withs=¹²₅, 3, p.

By Proposition 2.13, we know thatΦ(un)⇒Φ(u0)locally inR³. Since

∇u_n· ∇φ+ m²−

ω+Φ(u_n)2 u_nφ=

|u_n|^p−2u_nφ, ∀φ∈C₀^∞ R³

, letn→ ∞,

∇u₀· ∇φ+ m²−

ω+Φ(u₀)2 u₀φ=

|u₀|^p−2u₀φ.

That is,u₀is a weak radial solution of (3.6). Regarding (3.10) andL^p convergence of{u_n}, we imply thatu₀≡0.

Hence,u₀∈Σ. In addition, notice Remark 2.14. We have F J[u0] =

1

2|∇u0|²+1 2

m²−ω² u²₀−1

p|u0|^p−Eu₀

Φ(u0)

lim inf

n

1

2|∇u_n|²+1 2

m²−ω²

u²_n−lim

n

1

p|u_n|^p−lim

n E_un Φ(u_n)

=F.

i.e.F can be attained by the functionu₀∈Σ. 2 3.3. Solitary wave solutions – bounded smooth domain

In this section,U is a bounded smooth domain. We are interested in the existence of the positive least-J-action critical point ofJ. Therefore, we study the functionalJ₊, which is defined onH₀¹(U ). Due to Remark 3.2,J₊isC¹ differentiable in the sense of Fréchet.

Follow the notations as in [26]. Givene0,e≡0,e∈H₀¹(U )withJ₊[e] =0, we define C:= inf

h∈Γ max

0t1J₊ h(t )

(3.12) whereΓ is the set of all continuous paths joining the origin and this givene. In addition, we define

M[v] :=sup

t0

g_v(t ), ∀v∈H₀¹(U ), v0, v≡0 (3.13)