Multiplicity and Concentration of Solutions for Kirchhoff Equations with Magnetic Field

Chao Ji; Vicenţiu D. Rădulescu

doi:10.1515/ans-2021-2130

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Multiplicity and Concentration of Solutions for Kirchhoff Equations with Magnetic Field

Chao Ji and Vicenţiu D. Rădulescu

Published/Copyright: May 18, 2021

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From the journal Advanced Nonlinear Studies Volume 21 Issue 3

Abstract

In this paper, we study the following nonlinear magnetic Kirchhoff equation:

{ - ( a ⁢ ϵ 2 + b ⁢ ϵ ⁢ [ u ] A / ϵ 2 ) ⁢ Δ A / ϵ ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( | u | 2 ) ⁢ u in ⁢ ℝ 3 , u ∈ H 1 ⁢ ( ℝ 3 , ℂ ) ,

where ϵ>0, a,b>0 are constants, V:ℝ3→ℝ and A:ℝ3→ℝ3 are continuous potentials, and ΔA⁢u is the magnetic Laplace operator. Under a local assumption on the potential V, by combining variational methods, a penalization technique and the Ljusternik–Schnirelmann theory, we prove multiplicity properties of solutions and concentration phenomena for ϵ small. In this problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.

Keywords: Kirchhoff Equation; Magnetic Field; Concentration; Multiple Solutions; Variational Methods

MSC 2010: 35J20; 35J60; 58E05

1 Introduction and Main Results

This paper is devoted to the qualitative analysis of solutions for the nonlinear magnetic Kirchhoff equation in ℝ3. We are concerned with the existence and multiplicity of solutions, as well as with concentration properties of solutions for small values of the positive parameter. A feature of this paper is that the reaction has weak regularity, which allows to consider larger classes of nonlinearities. The main result is described in the final part of this section.

In this paper, we study the following nonlinear magnetic Kirchhoff equation:

(1.1) { - ( a ⁢ ϵ 2 + b ⁢ ϵ ⁢ [ u ] A / ϵ 2 ) ⁢ Δ A / ϵ ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( | u | 2 ) ⁢ u in ⁢ ℝ 3 , u ∈ H 1 ⁢ ( ℝ 3 , ℂ ) ,

where ϵ>0, a,b>0 are constants, V:ℝ3→ℝ is a continuous function, the magnetic potential A:ℝ3→ℝ3 is Hölder continuous with exponent α∈(0,1], and -ΔA⁢u is the magnetic Laplace operator of the following form:

- Δ A ⁢ u := ( 1 i ⁢ ∇ - A ⁢ ( x ) ) 2 ⁢ u = - Δ ⁢ u - 2 i ⁢ A ⁢ ( x ) ⋅ ∇ ⁡ u + | A ⁢ ( x ) | 2 ⁢ u - 1 i ⁢ u ⁢ div ⁡ ( A ⁢ ( x ) ) .

The definition of [u]A/ϵ2 will be given in Section 2.

For problem (1.1), there is a vast literature concerning the existence and multiplicity of bound state solutions for the case A≡0 and a=b=0. The first result in this direction was given by Floer and Weinstein in [8], where the case N=1 and f=iℝ is considered. Later on, several authors generalized this result to larger values of N, using different methods. For instance, He and Zou [10] considered the following fractional Schrödinger equation:

ϵ 2 ⁢ s ⁢ ( - Δ ) s ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( u ) + u 2 s * - 1 , x ∈ ℝ N ,

where V is a positive continuous function and satisfies the local assumption infx∈Λ⁡V⁢(x)<minx∈∂⁡Λ⁡V⁢(x), and f∈C is a function having subcritical and superlinear growth. By using the Nehari manifold method and the Ljusternik–Schnirelmann category theory, they obtained the multiplicity of positive solutions. We note that f is only continuous, and the Nehari manifold is only a topological manifold. He and Zou [10] applied the method that Szulkin and Weth developed in [20]. He and Zou [11] also studied multiplicity of concentrating solutions for a class of fractional Kirchhoff equations when the potential satisfies a local assumption and the nonlinear term f is only continuous. We also note that Ji, Fang and Zhang [12] considered a multiplicity result for asymptotically linear Kirchhoff equations. For further results about Kirchhoff equations, see [9, 19, 22, 23] and the references therein.

On the other hand, when a=b=0, the magnetic nonlinear Schrödinger equation (1.1) has been extensively investigated by many authors applying suitable variational and topological methods (see [1, 3, 4, 24, 7, 16, 15, 18, 25] and the references therein). It is well known that the first result involving the magnetic field was obtained by Esteban and Lions [7]. They used the concentration-compactness principle and minimization arguments to obtain solutions for ϵ>0 fixed and N=2,3. In particular, due to our scope, we want to mention [1] where Alves, Figueiredo and Furtado used the method of the Nehari manifold, the penalization method and Ljusternik–Schnirelmann category theory for subcritical nonlinearity f∈C1. We point out that if f is only continuous, then the arguments developed in [1] fail. In [13, 14], Ji and Rădulescu used the method of the Nehari manifold, the penalization method and Ljusternik–Schnirelmann category theory to study the multiplicity and concentration results for a magnetic Schrödinger equation in which the nonlinearity f is only continuous and subcritical and critical nonlinear terms, respectively. We also note the recent contribution [2] where Ambrosio studied multiplicity and concentration of solutions for a fractional Kirchhoff equation with magnetic field and critical growth.

Motivated by [11, 13], in the present paper, our main goal is to study multiplicity and concentration of nontrivial solutions for problem (1.1) only when f is continuous. Comparing with the result in [13], due to the presence of the nonlocal term, it is not clear to show the weak convergence of a bounded (PS) sequence of problem (1.1) is a solution of problem (1.1). Moreover, as we will see later, due to the presence of the magnetic field A⁢(x), problem (1.1) cannot be changed into a pure real-valued problem, and hence we should deal with a complex-valued problem directly, which causes several new difficulties in employing the methods in dealing with our problem. Our problem is more complicated than the pattern without magnetic field and we need additional technical estimates.

Throughout the paper, we make the following assumptions on the potential V:

There exists V0>0 such that V⁢(x)≥V0 for all x∈ℝ3.
There exists a bounded open set Λ⊂ℝ3 such that

V 0 = min x ∈ Λ ⁡ V ⁢ ( x ) < min x ∈ ∂ ⁡ Λ ⁡ V ⁢ ( x ) .

Observe that

M := { x ∈ Λ : V ⁢ ( x ) = V 0 } ≠ ∅ .

Moreover, let the nonlinearity f∈C⁢(ℝ,ℝ) be a function satisfying the following conditions:

f ⁢ ( t ) = 0 if t≤0, and limt→0+⁡f⁢(t)t=0.
There exists q∈(4,6) such that

lim t → + ∞ ⁡ f ⁢ ( t ) t q - 2 2 = 0 .
There is a positive constant θ>4 such that

0 < θ 2 ⁢ F ⁢ ( t ) ≤ t ⁢ f ⁢ ( t ) for all ⁢ t > 0 , where ⁢ F ⁢ ( t ) = ∫ 0 t f ⁢ ( s ) ⁢ 𝑑 s .
f ⁢ ( t ) t is strictly increasing in (0,∞).

The main result of this paper is the following theorem.

Theorem 1.1.

Assume that V satisfies (V1), (V2) and f satisfies (f1)–(f4). Then, for any δ>0 such that

M δ := { x ∈ ℝ 3 : dist ⁡ ( x , M ) < δ } ⊂ Λ ,

there exists ϵδ>0 such that, for any 0<ϵ<ϵδ, problem (1.1) has at least catMδ⁡(M) nontrivial solutions. Moreover, for every sequence {ϵn} such that ϵn→0+ as n→+∞, if we denote by uϵn one of these solutions of problem (1.1) for ϵ=ϵn and if ηϵn∈R3 is the global maximum point of |uϵn|, then

lim ϵ n → 0 + ⁡ V ⁢ ( η ϵ n ) = V 0 .

The paper is organized as follows. In Section 2, we introduce the functional setting and give some preliminaries. In Section 3, we study the modified problem and prove the Palais–Smale condition for the modified functional, and provide some tools which are useful to establish a multiplicity result. In Section 4, we study the autonomous problem associated. It allows us to show that the modified problem has multiple solutions. Finally, in Section 5, we give the proof of Theorem 1.1.

Notation.

C , C 1 , C 2 , … denote positive constants whose exact values are inessential and can change from line to line.
B R ⁢ ( y ) denotes the open ball centered at y∈ℝ3 with radius R>0, and BRc⁢(y) denotes the complement of BR⁢(y) in ℝ3.
∥ ⋅ ∥ , ∥⋅∥q and ∥⋅∥L∞⁢(Ω) denote the usual norms of the spaces H1⁢(ℝ3,ℝ), Lq⁢(ℝ3,ℝ) and L∞⁢(Ω,ℝ), respectively, where Ω⊂ℝ3.

2 Abstract Setting

In this section, we introduce the function spaces and some useful preliminary remarks, which will be useful for our arguments.

For u:ℝ3→ℂ, we set

∇ A ⁡ u := ( ∇ i - A ) ⁢ u .

Consider the function spaces

D A 1 ⁢ ( ℝ 3 , ℂ ) := { u ∈ L 6 ⁢ ( ℝ 3 , ℂ ) : | ∇ A ⁡ u | ∈ L 2 ⁢ ( ℝ 3 , ℝ ) }

and

H A 1 ⁢ ( ℝ 3 , ℂ ) := { u ∈ D A 1 ⁢ ( ℝ 3 , ℂ ) : u ∈ L 2 ⁢ ( ℝ 3 , ℂ ) } .

The space HA1⁢(ℝ3,ℂ) is a Hilbert space endowed with the scalar product

〈 u , v 〉 := Re ⁢ ∫ ℝ 3 ( ∇ A ⁡ u ⁢ ∇ A ⁡ v ¯ + u ⁢ v ¯ ) ⁢ 𝑑 x for any ⁢ u , v ∈ H A 1 ⁢ ( ℝ 3 , ℂ ) ,

where Re and the bar denote the real part of a complex number and the complex conjugation, respectively. We denote by ∥u∥A the norm induced by this inner product, and [u]A2:=∫ℝ3|∇A⁡u|2⁢𝑑x.

On HA1⁢(ℝ3,ℂ) we will frequently use the following diamagnetic inequality (see, e.g., [17, Theorem 7.21]):

(2.1) | ∇ A ⁡ u ⁢ ( x ) | ≥ | ∇ ⁡ | u ⁢ ( x ) | | for all ⁢ u ∈ H A 1 ⁢ ( ℝ 3 , ℂ ) .

Moreover, making a simple change of variables, since

Δ A ϵ = ϵ 2 ⁢ Δ A / ϵ and [ u ] A ϵ 2 = 1 ϵ ⁢ [ u ] A / ϵ 2 ,

we can see that problem (1.1) is equivalent to

(2.2) - ( a + b ⁢ [ u ] A ϵ 2 ) ⁢ Δ A ϵ ⁢ u + V ϵ ⁢ ( x ) ⁢ u = f ⁢ ( | u | 2 ) ⁢ u in ⁢ ℝ 3 ,

where Aϵ⁢(x)=A⁢(ϵ⁢x) and Vϵ⁢(x)=V⁢(ϵ⁢x).

Let Hϵ be the Hilbert space obtained as the closure of Cc∞⁢(ℝN,ℂ) with respect to the scalar product

〈 u , v 〉 ϵ := Re ⁢ ∫ ℝ 3 ( ∇ A ϵ ⁡ u ⁢ ∇ A ϵ ⁡ v ¯ + V ϵ ⁢ ( x ) ⁢ u ⁢ v ¯ ) ⁢ 𝑑 x

and let ∥⋅∥ϵ denote the norm induced by this inner product.

The diamagnetic inequality (2.1) implies that, if u∈HAϵ1⁢(ℝ3,ℂ), then |u|∈H1⁢(ℝ3,ℝ) and ∥u∥≤C⁢∥u∥ϵ. Therefore, the embedding Hϵ↪Lr⁢(ℝ3,ℂ) is continuous for 2≤r≤6 and the embedding Hϵ↪Llocr⁢(ℝ3,ℂ) is compact for 1≤r<6.

3 The Modified Problem

As in [6], to study problem (1.1), or equivalently (2.2), we modify suitably the nonlinearity f so that, for ϵ>0 small enough, the solutions of such modified problem are also solutions of the original one. More precisely, we choose K>2. By (f4), there exists a unique number a>0 verifying K⁢f⁢(a0)=V0, where V0 is given in (V1). Hence we consider the function

f ~ ⁢ ( t ) := { f ⁢ ( t ) , t ≤ a 0 , V 0 K , t > a 0 .

Now we introduce the penalized nonlinearity g:ℝ3×ℝ→ℝ:

g ⁢ ( x , t ) := χ Λ ⁢ ( x ) ⁢ f ⁢ ( t ) + ( 1 - χ Λ ⁢ ( x ) ) ⁢ f ~ ⁢ ( t ) ,

where χΛ is the characteristic function on Λ. Set G⁢(x,t):=∫0tg⁢(x,s)⁢𝑑s.

In view of (f1)–(f4), we have that g is a Carathéodory function satisfying the following properties:

g ⁢ ( x , t ) = 0 for each t≤0.
lim t → 0 + ⁢ g ⁢ ( x , t ) t = 0 uniformly in x∈ℝ3, and there exists q∈(4,6) such that

lim t → + ∞ ⁡ g ⁢ ( x , t ) t q - 2 2 = 0 uniformly in ⁢ x ∈ ℝ 3 .
g ⁢ ( x , t ) ≤ f ⁢ ( t ) for all t≥0 and uniformly in x∈ℝ3.
0 < θ ⁢ G ⁢ ( x , t ) ≤ 2 ⁢ g ⁢ ( x , t ) ⁢ t for each x∈Λ, t>0.
0 < G ⁢ ( x , t ) ≤ g ⁢ ( x , t ) ⁢ t ≤ V 0 ⁢ t / K for each x∈Λc, t>0.
For each x∈Λ, the function t↦g⁢(x,t)t is strictly increasing in t∈(0,+∞), and for each x∈Λc the function t↦g⁢(x,t)t is strictly increasing in (0,a0).

Next, we consider the modified problem

(3.1) - ( a + b ⁢ [ u ] A ϵ 2 ) ⁢ Δ A ϵ ⁢ u + V ϵ ⁢ ( x ) ⁢ u = g ⁢ ( ϵ ⁢ x , | u | 2 ) ⁢ u in ⁢ ℝ 3 .

Note that, if u is a solution of problem (3.1) with

| u ⁢ ( x ) | 2 ≤ a 0 for all ⁢ x ∈ Λ ϵ c , Λ ϵ := { x ∈ ℝ 3 : ϵ ⁢ x ∈ Λ } ,

then u is a solution of problem (2.2).

The functional associated to (3.1) is

J ϵ ⁢ ( u ) := a 2 ⁢ [ u ] A ϵ 2 + 1 2 ⁢ ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u | 2 ⁢ 𝑑 x + b 4 ⁢ [ u ] A ϵ 4 - 1 2 ⁢ ∫ ℝ 3 G ⁢ ( ϵ ⁢ x , | u | 2 ) ⁢ 𝑑 x

defined in Hϵ. It is standard to prove that Jϵ∈C1⁢(Hϵ,ℝ) and its critical points are the weak solutions of the modified problem (3.1).

We denote by 𝒩ϵ the Nehari manifold of Jϵ, that is,

𝒩 ϵ := { u ∈ H ϵ ∖ { 0 } : J ϵ ′ ⁢ ( u ) ⁢ [ u ] = 0 } ,

and define the number cϵ by

c ϵ = inf u ∈ 𝒩 ϵ ⁡ J ϵ ⁢ ( u ) .

Let Hϵ+ be the open subset Hϵ given by

H ϵ + = { u ∈ H ϵ : | supp ⁡ ( u ) ∩ Λ ϵ | > 0 } ,

and Sϵ+=Sϵ∩Hϵ+, where Sϵ is the unit sphere of Hϵ. Note that Sϵ+ is a non-complete C1,1-manifold of codimension 1, modeled on Hϵ and contained in Hϵ+. Therefore, Hϵ=Tu⁢Sϵ+⁢⊕ℝ⁢u for each u∈Tu⁢Sϵ+, where Tu⁢Sϵ+={v∈Hϵ:〈u,v〉ϵ=0}.

Now we show that the functional Jϵ satisfies the mountain pass geometry.

Lemma 3.1.

For any fixed ϵ>0, the functional Jϵ satisfies the following properties:

There exist β , r > 0 such that J ϵ ⁢ ( u ) ≥ β if ∥ u ∥ ϵ = r .
There exists e ∈ H ϵ with ∥ e ∥ ϵ > r such that J ϵ ⁢ ( e ) < 0 .

Proof.

(i) By (g2), (g4) and (g5), for any ζ>0 small, there exists Cζ>0 such that

G ⁢ ( ϵ ⁢ x , | u | 2 ) ≤ ζ ⁢ | u | 4 + C ζ ⁢ | u | q for all ⁢ x ∈ ℝ 3 .

By the Sobolev embedding theorem, it follows that

J ϵ ⁢ ( u ) ≥ a 2 ⁢ [ u ] A ϵ 2 + 1 2 ⁢ ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u | 2 ⁢ 𝑑 x + b 4 ⁢ [ u ] A ϵ 4 - ζ 2 ⁢ ∫ ℝ 3 | u | 4 ⁢ 𝑑 x - C ζ 2 ⁢ ∫ ℝ 3 | u | q ⁢ 𝑑 x

≥ 1 2 ⁢ ∥ u n ∥ ϵ 2 - C 1 ⁢ ζ ⁢ ∥ u n ∥ ϵ 4 - C 2 ⁢ C ζ ⁢ ∥ u n ∥ ϵ q .

Hence we can choose some β,r>0 such that Jϵ⁢(u)≥β if ∥u∥ϵ=r since q>4.

(ii) For each u∈Hϵ+ and t>0, by the definition of g and (f3), one has

J ϵ ⁢ ( t ⁢ u ) ≤ t 2 2 ⁢ ∥ u ∥ ϵ 2 + b ⁢ t 4 4 ⁢ [ u ] A ϵ 4 - 1 2 ⁢ ∫ Λ ϵ G ⁢ ( ϵ ⁢ x , t 2 ⁢ | u | 2 ) ⁢ 𝑑 x

≤ t 2 2 ⁢ ∥ u ∥ ϵ 2 + b ⁢ t 4 4 ⁢ [ u ] A ϵ 4 - C 1 ⁢ t θ ⁢ ∫ Λ ϵ | u | θ ⁢ 𝑑 x + C 2 ⁢ | supp ⁡ ( u ) ∩ Λ ϵ | .

Since θ>4, we can get the conclusion. ∎

Since f is only continuous, the next results are very important because they allow us to overcome the non-differentiability of 𝒩ϵ and the incompleteness of Sϵ+.

Lemma 3.2.

Assume that (V1)–(V2) and (f1)–(f4) are satisfied. Then the following properties hold:

For any u ∈ H ϵ + , let g u : ℝ + → ℝ be given by g u ⁢ ( t ) = J ϵ ⁢ ( t ⁢ u ) . Then there exists a unique t u > 0 such that g u ′ ⁢ ( t ) > 0 in ( 0 , t u ) and g u ′ ⁢ ( t ) < 0 in ( t u , ∞ ) .
There is some τ > 0 independent of u such that t u ≥ τ for all u ∈ S ϵ + . Moreover, for each compact 𝒲 ⊂ S ϵ + there is a constant C 𝒲 such that t u ≤ C 𝒲 for all u ∈ 𝒲 .
The map m ^ ϵ : H ϵ + → 𝒩 ϵ given by m ^ ϵ ⁢ ( u ) = t u ⁢ u is continuous, and m ϵ = m ^ ϵ | S ϵ + is a homeomorphism between S ϵ + and 𝒩 ϵ . Moreover, m ϵ - 1 ⁢ ( u ) = u ∥ u ∥ ϵ .
If there is a sequence { u n } ⊂ S ϵ + such that dist ⁡ ( u n , ∂ ⁡ S ϵ + ) → 0 , then ∥ m ϵ ⁢ ( u n ) ∥ ϵ → ∞ and J ϵ ⁢ ( m ϵ ⁢ ( u n ) ) → ∞ .

Proof.

(A1) As in the proof of Lemma 3.1, we have gu⁢(0)=0, gu⁢(t)>0 for t>0 small, and gu⁢(t)<0 for t>0 large. Therefore, maxt≥0⁡gu⁢(t) is achieved at a global maximum point t=tu verifying gu′⁢(tu)=0 and tu⁢u∈𝒩ϵ. Now, we show that tu is unique. Arguing by contradiction, suppose that there exist t1>t2>0 such that gu′⁢(t1)=gu′⁢(t2)=0. Then, for i=1,2,

t i ⁢ a ⁢ [ u ] A ϵ 2 + t i ⁢ ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u | 2 ⁢ 𝑑 x + t i 3 ⁢ b ⁢ [ u ] A ϵ 4 = ∫ ℝ 3 g ⁢ ( ϵ ⁢ x , t i 2 ⁢ | u | 2 ) ⁢ t i ⁢ | u | 2 ⁢ 𝑑 x .

Hence,

a ⁢ [ u ] A ϵ 2 + ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u | 2 ⁢ 𝑑 x t i 2 + b ⁢ [ u ] A ϵ 4 = ∫ ℝ 3 g ⁢ ( ϵ ⁢ x , t i 2 ⁢ | u | 2 ) ⁢ | u | 2 t i 2 ⁢ 𝑑 x ,

which implies that

( 1 t 1 2 - 1 t 2 2 ) ⁢ ( a ⁢ [ u ] A ϵ 2 + ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u | 2 ⁢ 𝑑 x )

= ∫ ℝ 3 ( g ⁢ ( ϵ ⁢ x , t 1 2 ⁢ | u | 2 ) t 1 2 ⁢ | u | 2 - g ⁢ ( ϵ ⁢ x , t 2 2 ⁢ | u | 2 ) t 2 2 ⁢ | u | 2 ) ⁢ | u | 4 ⁢ 𝑑 x

≥ ∫ Λ ϵ c ∩ { t 2 2 | u | 2 ≤ a 0 ≤ t 1 2 | u | 2 } ( g ⁢ ( ϵ ⁢ x , t 1 2 ⁢ | u | 2 ) t 1 2 ⁢ | u | 2 - g ⁢ ( ϵ ⁢ x , t 2 2 ⁢ | u | 2 ) t 2 2 ⁢ | u | 2 ) ⁢ | u | 4 ⁢ 𝑑 x

+ ∫ Λ ϵ c ∩ { a 0 ≤ t 2 2 | u | 2 } ( g ⁢ ( ϵ ⁢ x , t 1 2 ⁢ | u | 2 ) t 1 2 ⁢ | u | 2 - g ⁢ ( ϵ ⁢ x , t 2 2 ⁢ | u | 2 ) t 2 2 ⁢ | u | 2 ) ⁢ | u | 4 ⁢ 𝑑 x

≥ ∫ Λ ϵ c ∩ { t 2 2 | u | 2 ≤ a 0 ≤ t 1 2 | u | 2 } ( V 0 K ⁢ 1 t 1 2 ⁢ | u | 2 - f ⁢ ( t 2 2 ⁢ | u | 2 ) t 2 2 ⁢ | u | 2 ) ⁢ | u | 4 ⁢ 𝑑 x + 1 K ⁢ ( 1 t 1 2 - 1 t 2 2 ) ⁢ ∫ Λ ϵ c ∩ { a 0 ≤ t 2 2 | u | 2 } V 0 ⁢ | u | 2 ⁢ 𝑑 x .

Since t1>t2>0, we have

( a ⁢ [ u ] A ϵ 2 + ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u | 2 ⁢ 𝑑 x )

≤ t 1 2 ⁢ t 2 2 t 2 2 - t 1 2 ⁢ ∫ Λ ϵ c ∩ { t 2 2 | u | 2 ≤ a 0 ≤ t 1 2 | u | 2 } ( V 0 K ⁢ 1 t 1 2 ⁢ | u | 2 - f ⁢ ( t 2 2 ⁢ | u | 2 ) t 2 2 ⁢ | u | 2 ) ⁢ | u | 4 ⁢ 𝑑 x + 1 K ⁢ ∫ Λ ϵ c ∩ { a 0 ≤ t 2 2 | u | 2 } V 0 ⁢ | u | 2 ⁢ 𝑑 x

≤ 1 K ⁢ ∫ Λ ϵ c V 0 ⁢ | u | 2 ⁢ 𝑑 x

≤ 1 K ⁢ ∥ u ∥ ϵ 2 ,

which is a contradiction. Therefore, maxt≥0⁡gu⁢(t) is achieved at a unique t=tu so that gu′⁢(t)=0 and tu⁢u∈𝒩ϵ.

(A2) For any u∈Sϵ+, we have

t u + t u 3 ⁢ b ⁢ [ u ] A ϵ 4 = ∫ ℝ 3 g ⁢ ( ϵ ⁢ x , t u 2 ⁢ | u | 2 ) ⁢ t u ⁢ | u | 2 ⁢ 𝑑 x .

From (g2), Sobolev embeddings and since q>4, we get

t u ≤ ζ ⁢ t u 3 ⁢ ∫ ℝ 3 | u | 4 ⁢ 𝑑 x + C ζ ⁢ t u q - 1 ⁢ ∫ ℝ 3 | u | q ⁢ 𝑑 x ≤ C 1 ⁢ ζ ⁢ t u 3 + C 2 ⁢ C ζ ⁢ t u q - 1 ,

which implies that tu≥τ for some τ>0. Suppose by contradiction that there is {un}⊂𝒲 with tn:=tun→∞. Since 𝒲 is compact, there exists u∈𝒲 such that un→u in Hϵ. Moreover, using the proof of Lemma 3.1 (ii), we have that Jϵ⁢(tn⁢un)→-∞.

On the other hand, let vn:=tn⁢un∈𝒩ϵ. From the definition of g and by (g4), (g5) and θ>4, it follows that

J ϵ ⁢ ( v n ) = J ϵ ⁢ ( v n ) - 1 θ ⁢ J ϵ ′ ⁢ ( v n ) ⁢ [ v n ]

≥ ( 1 2 - 1 θ ) ⁢ ∥ v n ∥ ϵ 2 + ( 1 4 - 1 θ ) ⁢ b ⁢ [ v n ] A ϵ 4 + ∫ Λ ϵ c ( 1 θ ⁢ g ⁢ ( ϵ ⁢ x , | v n | 2 ) ⁢ | v n | 2 - 1 2 ⁢ G ⁢ ( ϵ ⁢ x , | v n | 2 ) ) ⁢ 𝑑 x

≥ ( 1 2 - 1 θ ) ⁢ ( ∥ v n ∥ ϵ 2 - 1 K ⁢ ∫ ℝ 3 V ⁢ ( ϵ ⁢ x ) ⁢ | v n | 2 ⁢ 𝑑 x )

≥ ( 1 2 - 1 θ ) ⁢ ( 1 - 1 K ) ⁢ ∥ v n ∥ ϵ 2 .

Thus, substituting vn:=tn⁢un and ∥vn∥ϵ=tn, we obtain

0 < ( 1 2 - 1 θ ) ⁢ ( 1 - 1 K ) ≤ J ϵ ⁢ ( v n ) t n 2 ≤ 0

as n→∞, which yields a contradiction. This proves (A2).

(A3) First of all, we note that m^ϵ, mϵ and mϵ-1 are well defined. Indeed, by (A2), for each u∈Hϵ+, there is a unique m^ϵ⁢(u)∈𝒩ϵ. On the other hand, if u∈𝒩ϵ, then u∈Hϵ+. Otherwise, we have |supp⁡(u)∩Λϵ|=0 and by (g5), we have

a ⁢ [ u ] A ϵ 2 + ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u | 2 ⁢ 𝑑 x + b ⁢ [ u ] A ϵ 4 = ∫ ℝ 3 g ⁢ ( ϵ ⁢ x , | u | 2 ) ⁢ | u | 2 ⁢ 𝑑 x

= ∫ Λ ϵ c g ⁢ ( ϵ ⁢ x , | u | 2 ) ⁢ | u | 2 ⁢ 𝑑 x

≤ 1 K ⁢ ∫ ℝ 3 V ⁢ ( ϵ ⁢ x ) ⁢ | u | 2 ⁢ 𝑑 x

≤ 1 K ⁢ ∥ u ∥ ϵ 2 ,

which is impossible since K>2 and u≠0. Therefore, mϵ-1⁢(u)=u∥u∥ϵ∈Sϵ+ is well defined and continuous. From

m ϵ - 1 ⁢ ( m ϵ ⁢ ( u ) ) = m ϵ - 1 ⁢ ( t u ⁢ u ) = t u ⁢ u t u ⁢ ∥ u ∥ ϵ = u for all ⁢ u ∈ S ϵ + ,

we conclude that mϵ is a bijection.

Now we prove that m^ϵ:Hϵ+→𝒩ϵ is continuous. Let {un}⊂Hϵ+ and u∈Hϵ+ such that un→u in Hϵ. By (A2), there exists t0>0 such that tn:=tun→t0. Using tn⁢un∈𝒩ϵ, that is,

t n 2 ⁢ a ⁢ [ u n ] A ϵ 2 + t n 2 ⁢ ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u n | 2 ⁢ 𝑑 x + t n 4 ⁢ b ⁢ [ u n ] A ϵ 4 = ∫ ℝ 3 g ⁢ ( ϵ ⁢ x , t n 2 ⁢ | u n | 2 ) ⁢ t n 2 ⁢ | u n | 2 ⁢ 𝑑 x for all ⁢ n ∈ N ,

and passing to the limit as n→∞ in the last inequality, we obtain

t 0 2 ⁢ a ⁢ [ u ] A ϵ 2 + t 0 2 ⁢ ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u | 2 ⁢ 𝑑 x + t 0 4 ⁢ b ⁢ [ u ] A ϵ 4 = ∫ ℝ 3 g ⁢ ( ϵ ⁢ x , t 0 2 ⁢ | u | 2 ) ⁢ t 0 2 ⁢ | u | 2 ⁢ 𝑑 x ,

which implies that t0⁢u∈𝒩ϵ and tu=t0. This proves that m^ϵ⁢(un)→m^ϵ⁢(u) in Hϵ+. Thus, m^ϵ and mϵ are continuous functions and (A3) is proved.

(A4) Let {un}⊂Sϵ+ be a subsequence such that dist⁡(un,∂⁡Sϵ+)→0. Then, for each v∈∂⁡Sϵ+ and n∈N, we have |un|=|un-v| a.e. in Λϵ. Therefore, by (V1), (V2) and the Sobolev embedding theorem, there exists a constant Cr>0 such that

∥ u n ∥ L r ⁢ ( Λ ϵ ) ≤ inf v ∈ ∂ ⁡ S ϵ + ⁡ ∥ u n - v ∥ L r ⁢ ( Λ ϵ )

≤ C r ⁢ ( inf v ∈ ∂ ⁡ S ϵ + ⁡ ∫ Λ ϵ ( | ∇ A ϵ ⁡ u n - v | 2 + V ϵ ⁢ ( x ) ⁢ | u n - v | 2 ) ⁢ 𝑑 x ) 1 2

≤ C r ⁢ dist ⁡ ( u n , ∂ ⁡ S ϵ + )

for all n∈N and r∈[2,6]. By (g2), (g3) and (g5), for each t>0, we have

∫ ℝ N G ⁢ ( ϵ ⁢ x , t 2 ⁢ | u n | 2 ) ⁢ 𝑑 x ≤ ∫ Λ ϵ F ⁢ ( t 2 ⁢ | u n | 2 ) ⁢ 𝑑 x + t 2 K ⁢ ∫ Λ ϵ c V ⁢ ( ϵ ⁢ x ) ⁢ | u n | 2 ⁢ 𝑑 x

≤ C 1 ⁢ t 4 ⁢ ∫ Λ ϵ | u n | 4 ⁢ 𝑑 x + C 2 ⁢ t q ⁢ ∫ Λ ϵ | u n | q ⁢ 𝑑 x + t 2 K ⁢ ∥ u n ∥ ϵ 2

≤ C 3 t 4 dist ( u n , ∂ S ϵ + ) 4 + C 4 t q dist ( u n , ∂ S ϵ + ) q + t 2 K .

Therefore,

lim sup n ⁡ ∫ ℝ 3 G ⁢ ( ϵ ⁢ x , t 2 ⁢ | u n | 2 ) ⁢ 𝑑 x ≤ t 2 K for all ⁢ t > 0 .

On the other hand, from the definition of mϵ and the last inequality, for all t>0, one has

lim inf n ⁡ J ϵ ⁢ ( m ϵ ⁢ ( u n ) ) ≥ lim inf n ⁡ J ϵ ⁢ ( t ⁢ u n )

≥ lim ⁡ inf n ⁡ t 2 2 ⁢ ∥ u n ∥ ϵ 2 - t 2 K

= K - 2 2 ⁢ K ⁢ t 2 .

This implies that

lim inf n ⁡ 1 2 ⁢ ∥ m ϵ ⁢ ( u n ) ∥ ϵ 2 ≥ K - 2 2 ⁢ K ⁢ t 2 for all ⁢ t > 0 .

From the arbitrariness of t>0, it is easy to see that ∥mϵ⁢(un)∥ϵ→∞ and Jϵ⁢(mϵ⁢(un))→∞ as n→∞. This completes the proof of Lemma 3.2. ∎

Now we define the function

Ψ ^ ϵ : H ϵ + → ℝ

by Ψ^ϵ⁢(u)=Jϵ⁢(m^ϵ⁢(u)) and set Ψϵ:=(Ψ^ϵ)|Sϵ+.

From Lemma 3.2, we have the following result.

Lemma 3.3.

Assume that (V1)–(V2) and (f1)–(f4) are satisfied. Then the following assertions hold:

Ψ ^ ϵ ∈ C 1 ⁢ ( H ϵ + , ℝ ) and

Ψ ^ ϵ ′ ⁢ ( u ) ⁢ v = ∥ m ^ ϵ ⁢ ( u ) ∥ ϵ ∥ u ∥ ϵ ⁢ J ϵ ′ ⁢ ( m ^ ϵ ⁢ ( u ) ) ⁢ [ v ] for all ⁢ u ∈ H ϵ + ⁢ and all ⁢ v ∈ H ϵ .
Ψ ϵ ∈ C 1 ⁢ ( S ϵ + , ℝ ) and

Ψ ϵ ′ ⁢ ( u ) ⁢ v = ∥ m ϵ ⁢ ( u ) ∥ ϵ ⁢ J ϵ ′ ⁢ ( m ^ ϵ ⁢ ( u ) ) ⁢ [ v ] for all ⁢ v ∈ T u ⁢ S ϵ + .
If { u n } is a ( PS ) c sequence of Ψ ϵ , then { m ϵ ⁢ ( u n ) } is a ( PS ) c sequence of J ϵ . If {un}⊂𝒩ϵ is a bounded (PS)c sequence of Jϵ, then {mϵ-1⁢(un)} is a (PS)c sequence of Ψϵ.
u is a critical point of Ψ ϵ if and only if m ϵ ⁢ ( u ) is a critical point of J ϵ . Moreover, the corresponding critical values coincide and

inf S ϵ + ⁡ Ψ ϵ = inf 𝒩 ϵ ⁡ J ϵ .

As in [21], we have the following variational characterization of the infimum of Jϵ over 𝒩ϵ:

c ϵ = inf u ∈ 𝒩 ϵ ⁡ J ϵ ⁢ ( u ) = inf u ∈ H ϵ + ⁡ sup t > 0 ⁡ J ϵ ⁢ ( t ⁢ u ) = inf u ∈ S ϵ + ⁡ sup t > 0 ⁡ J ϵ ⁢ ( t ⁢ u ) .

Lemma 3.4.

Let c>0 and let {un} be a (PS)c sequence for Jϵ. Then {un} is bounded in Hϵ.

Proof.

Assume that {un}⊂Hϵ is a (PS)c sequence for Jϵ, that is, Jϵ⁢(un)→c and Jϵ′⁢(un)→0. By using (g4), (g5) and θ>4, we have

c + o n ⁢ ( 1 ) + o n ⁢ ( 1 ) ⁢ ∥ u n ∥ ϵ ≥ J ϵ ⁢ ( u n ) - 1 θ ⁢ J ϵ ′ ⁢ ( u n ) ⁢ [ u n ]

= ( 1 2 - 1 θ ) ⁢ ∥ u n ∥ ϵ 2 + ( 1 4 - 1 θ ) ⁢ b ⁢ [ u n ] A ϵ 4 + ∫ ℝ 3 ( 1 θ ⁢ g ⁢ ( ϵ ⁢ x , | u n | 2 ) ⁢ | u n | 2 - 1 2 ⁢ G ⁢ ( ϵ ⁢ x , | u n | 2 ) ) ⁢ 𝑑 x

≥ ( 1 2 - 1 θ ) ⁢ ∥ u n ∥ ϵ 2 + ∫ Λ ϵ c ( 1 θ ⁢ g ⁢ ( ϵ ⁢ x , | u n | 2 ) ⁢ | u n | 2 - 1 2 ⁢ G ⁢ ( ϵ ⁢ x , | u n | 2 ) ) ⁢ 𝑑 x

≥ ( 1 2 - 1 θ ) ⁢ ∥ u n ∥ ϵ 2 - 1 2 ⁢ ∫ Λ ϵ c G ⁢ ( ϵ ⁢ x , | u n | 2 ) ⁢ 𝑑 x

≥ ( 1 2 - 1 θ ) ⁢ ∥ u n ∥ ϵ 2 - 1 2 ⁢ K ⁢ ∫ ℝ 3 V ⁢ ( ϵ ⁢ x ) ⁢ | u n | 2 ⁢ 𝑑 x

≥ ( 1 2 - 1 θ - 1 2 ⁢ K ) ⁢ ∥ u n ∥ ϵ 2 .

Since K>2, from the above inequalities we obtain that {un} is bounded in Hϵ. ∎

The following result is important to prove the (PS)cϵ condition for the functional Jϵ.

Lemma 3.5.

The functional Jϵ satisfies the (PS)c condition at any level c>0.

Proof.

Let (un)⊂Hϵ be a (PS)c sequence for Jϵ. By Lemma 3.4, (un) is bounded in Hϵ. Thus, up to a subsequence, un⇀u in Hϵ and un→u in Llocr⁢(ℝ3,ℂ) for all 1≤r<6 as n→+∞. Moreover, Jϵ′⁢(u)=0 and

a ⁢ [ u ] A ϵ 2 + ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u | 2 ⁢ 𝑑 x + b ⁢ [ u ] A ϵ 4 = ∫ ℝ 3 g ⁢ ( ϵ ⁢ x , | u | 2 ) ⁢ | u | 2 ⁢ 𝑑 x .

For the fixed ϵ>0, let R>0 be such that Λϵ⊂BR/2⁢(0). We show that for any given ζ>0, for R large enough,

(3.2) lim sup n ⁡ ∫ B R c ⁢ ( 0 ) ( | ∇ A ϵ ⁡ u n | 2 + V ϵ ⁢ ( x ) ⁢ | u n | 2 ) ⁢ 𝑑 x ≤ ζ .

Let ϕR∈C∞⁢(ℝ3,ℝ) be a cut-off function such that

ϕ R = 0 ⁢ for ⁢ x ∈ B R / 2 ⁢ ( 0 ) , ϕ R = 1 ⁢ for ⁢ x ∈ B R c ⁢ ( 0 ) , 0 ≤ ϕ R ≤ 1 , | ∇ ⁡ ϕ R | ≤ C R ,

where C>0 is a constant independent of R. Since the sequence (ϕR⁢un) is bounded in Hϵ, we have

J ϵ ′ ⁢ ( u n ) ⁢ [ ϕ R ⁢ u n ] = o n ⁢ ( 1 ) ,

that is

a ⁢ Re ⁢ ∫ ℝ 3 ∇ A ϵ ⁡ u n ⁢ ∇ A ϵ ⁡ ( ϕ R ⁢ u n ) ¯ ⁢ d ⁢ x + ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u n | 2 ⁢ ϕ R ⁢ 𝑑 x + b ⁢ [ u n ] A ϵ 2 ⁢ Re ⁢ ∫ ℝ 3 ∇ A ϵ ⁡ u n ⁢ ∇ A ϵ ⁡ ( ϕ R ⁢ u n ) ¯ ⁢ d ⁢ x

= ∫ ℝ 3 g ⁢ ( ϵ ⁢ x , | u n | 2 ) ⁢ | u n | 2 ⁢ ϕ R ⁢ 𝑑 x + o n ⁢ ( 1 ) .

Since

∇ A ϵ ⁡ ( u n ⁢ ϕ R ) ¯ = i ⁢ u n ¯ ⁢ ∇ ⁡ ϕ R + ϕ R ⁢ ∇ A ϵ ⁡ u n ¯ ,

using (g5), we have

∫ ℝ 3 ( a ⁢ | ∇ A ϵ ⁡ u n | 2 + V ϵ ⁢ ( x ) ⁢ | u n | 2 ) ⁢ ϕ R ⁢ 𝑑 x

≤ ∫ ℝ 3 g ⁢ ( ϵ ⁢ x , | u n | 2 ) ⁢ | u n | 2 ⁢ ϕ R ⁢ 𝑑 x - ( a + b ⁢ [ u n ] A ϵ 2 ) ⁢ Re ⁢ ∫ ℝ 3 i ⁢ u n ¯ ⁢ ∇ A ϵ ⁡ u n ⁢ ∇ ⁡ ϕ R ⁢ d ⁢ x + o n ⁢ ( 1 )

≤ 1 K ⁢ ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u n | 2 ⁢ ϕ R ⁢ 𝑑 x + C ⁢ | Re ⁢ ∫ ℝ 3 i ⁢ u n ¯ ⁢ ∇ A ϵ ⁡ u n ⁢ ∇ ⁡ ϕ R ⁢ d ⁢ x | + o n ⁢ ( 1 ) .

By the definition of ϕR, the Hölder inequality and the boundedness of (un) in Hϵ, we obtain

( 1 - 1 K ) ⁢ ∫ ℝ 3 ( a ⁢ | ∇ A ϵ ⁡ u n | 2 + V ϵ ⁢ ( x ) ⁢ | u n | 2 ) ⁢ ϕ R ⁢ 𝑑 x ≤ C R ⁢ ∥ u n ∥ 2 ⁢ ∥ ∇ A ϵ ⁡ u n ∥ 2 + o n ⁢ ( 1 ) ≤ C 1 R + o n ⁢ ( 1 ) ,

and so (3.2) holds.

Now, we prove that for any R>0 the following limit holds:

(3.3) lim sup n ⁡ ∫ B R ⁢ ( 0 ) ( | ∇ A ϵ ⁡ u n | 2 + V ϵ ⁢ ( x ) ⁢ | u n | 2 ) ⁢ 𝑑 x = ∫ B R ⁢ ( 0 ) ( | ∇ A ϵ ⁡ u | 2 + V ϵ ⁢ ( x ) ⁢ | u | 2 ) ⁢ 𝑑 x .

Let ϕρ∈C∞⁢(ℝ3,ℝ) be a cut-off function such that

ϕ ρ = 1 ⁢ for ⁢ x ∈ B ρ ⁢ ( 0 ) , ϕ ρ = 0 ⁢ for ⁢ x ∈ B 2 ⁢ ρ c ⁢ ( 0 ) , 0 ≤ ϕ ρ ≤ 1 , | ∇ ⁡ ϕ ρ | ≤ C ρ ,

where C>0 is a constant independent of ρ. Let

P n ⁢ ( x ) = M ⁢ ( u n ) ⁢ | ∇ A ϵ ⁡ u n - ∇ A ϵ ⁡ u | 2 + V ϵ ⁢ ( x ) ⁢ | u n - u | 2 ,

where

M ⁢ ( u n ) = a + b ⁢ ∫ ℝ 3 | ∇ A ϵ ⁡ u n | 2 ⁢ 𝑑 x .

For the fixed R>0, choosing ρ>R>0, we have

∫ B R P n ⁢ ( x ) ⁢ 𝑑 x ≤ ∫ ℝ 3 P n ⁢ ( x ) ⁢ ϕ ρ ⁢ ( x ) ⁢ 𝑑 x

= M ⁢ ( u n ) ⁢ ∫ ℝ 3 | ∇ A ϵ ⁡ u n - ∇ A ϵ ⁡ u | 2 ⁢ ϕ ρ ⁢ ( x ) ⁢ 𝑑 x + ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u n - u | 2 ⁢ ϕ ρ ⁢ ( x ) ⁢ 𝑑 x

(3.4) = J n , ρ 1 - J n , ρ 2 + J n , ρ 3 + J n , ρ 4 ,

where

J n , ρ 1 = M ⁢ ( u n ) ⁢ ∫ ℝ 3 | ∇ A ϵ ⁡ u n | 2 ⁢ ϕ ρ ⁢ ( x ) ⁢ 𝑑 x + ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ | u n | 2 ⁢ ϕ ρ ⁢ ( x ) ⁢ 𝑑 x - ∫ ℝ 3 g ⁢ ( ϵ ⁢ x , | u n | 2 ) ⁢ | u n | 2 ⁢ ϕ ρ ⁢ 𝑑 x ,

J n , ρ 2 = M ⁢ ( u n ) ⁢ Re ⁢ ∫ ℝ 3 ∇ A ϵ ⁡ u n ⁢ ∇ A ϵ ⁡ u ¯ ⁢ ϕ ρ ⁢ ( x ) ⁢ 𝑑 x + Re ⁢ ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ u n ⁢ u ¯ ⁢ ϕ ρ ⁢ ( x ) ⁢ 𝑑 x - Re ⁢ ∫ ℝ 3 g ⁢ ( ϵ ⁢ x , | u n | 2 ) ⁢ u n ⁢ u ¯ ⁢ ϕ ρ ⁢ ( x ) ⁢ 𝑑 x ,

J n , ρ 3 = - M ⁢ ( u n ) ⁢ Re ⁢ ∫ ℝ 3 ( ∇ A ϵ ⁡ u n - ∇ A ϵ ⁡ u ) ⁢ ∇ A ϵ ⁡ u ¯ ⁢ ϕ ρ ⁢ ( x ) ⁢ 𝑑 x + Re ⁢ ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ ( u n - u ) ⁢ u ¯ ⁢ ϕ ρ ⁢ ( x ) ⁢ 𝑑 x ,

J n , ρ 4 = Re ⁢ ∫ ℝ 3 g ⁢ ( ϵ ⁢ x , | u n | 2 ) ⁢ u n ⁢ ( u n - u ) ¯ ⁢ ϕ ρ ⁢ ( x ) ⁢ 𝑑 x .

It is easy to see that

J n , ρ 1 = J ϵ ′ ⁢ ( u n ) ⁢ [ ϕ ρ ⁢ u n ] - M ⁢ ( u n ) ⁢ Re ⁢ ∫ ℝ 3 i ⁢ u n ¯ ⁢ ∇ A ϵ ⁡ u n ⁢ ∇ ⁡ ϕ ρ ⁢ d ⁢ x

and

J n , ρ 2 = J ϵ ′ ⁢ ( u n ) ⁢ [ ϕ ρ ⁢ u ] - M ⁢ ( u n ) ⁢ Re ⁢ ∫ ℝ 3 i ⁢ u ¯ ⁢ ∇ A ϵ ⁡ u n ⁢ ∇ ⁡ ϕ ρ ⁢ d ⁢ x .

Then

lim ρ → ∞ ⁢ lim ⁡ sup n → ∞ ⁢ | J n , ρ 1 | = 0 , lim ρ → ∞ ⁢ lim ⁡ sup n → ∞ ⁢ | J n , ρ 2 | = 0 .

On the other hand, since the sequence (un) is bounded in Hϵ, we assume that

∫ ℝ 3 | ∇ A ϵ ⁡ u n | 2 ⁢ 𝑑 x → l 2 .

Then

J n , ρ 3 = - ( a + b ⁢ l 2 ) ⁢ Re ⁢ ∫ ℝ 3 ( ∇ A ϵ ⁡ u n - ∇ A ϵ ⁡ u ) ⁢ ∇ A ϵ ⁡ ( u ⁢ ϕ ρ ⁢ ( x ) ) ¯ ⁢ 𝑑 x - Re ⁢ ∫ ℝ 3 V ϵ ⁢ ( x ) ⁢ ( u n - u ) ⁢ ( u ⁢ ϕ ρ ⁢ ( x ) ) ¯ ⁢ 𝑑 x

+ ( a + b ⁢ l 2 ) ⁢ Re ⁢ ∫ ℝ 3 ( ∇ A ϵ ⁡ u n - ∇ A ϵ ⁡ u ) ⁢ i ⁢ u ¯ ⁢ ∇ ⁡ ϕ ρ ⁢ d ⁢ x + o n ⁢ ( 1 )

= - ( a + b ⁢ l 2 ) ⁢ 〈 u n - u , u ⁢ ϕ ρ ⁢ ( x ) 〉 + ( a + b ⁢ l 2 ) ⁢ Re ⁢ ∫ ℝ 3 ( ∇ A ϵ ⁡ u n - ∇ A ϵ ⁡ u ) ⁢ i ⁢ u ¯ ⁢ ∇ ⁡ ϕ ρ ⁢ d ⁢ x + o n ⁢ ( 1 ) ,

and thus

lim ρ → ∞ ⁢ lim ⁡ sup n → ∞ ⁢ | J n , ρ 3 | = 0 .

Now we prove that

(3.5) lim ρ → ∞ ⁢ lim ⁡ sup n → ∞ ⁢ | J n , ρ 4 | = 0 .

It is easy to see that

J n , ρ 4 ≤ ∫ ( ℝ 3 ∖ Λ ϵ ) ∩ B 2 ⁢ ρ ⁢ ( 0 ) | g ( ϵ x , | u n | 2 ) u n ( u n - u ) ¯ | d x + ∫ Λ ϵ ∩ B 2 ⁢ ρ ⁢ ( 0 ) | g ( ϵ x , | u n | 2 ) u n ( u n - u ) ¯ | d x .

Using the Sobolev compact embedding Hϵ↪Llocr⁢(ℝ3,ℂ) for 1≤r<6, (g5), (f1) and (f2) imply that

∫ ( ℝ 3 ∖ Λ ϵ ) ∩ B 2 ⁢ ρ ⁢ ( 0 ) | g ⁢ ( ϵ ⁢ x , | u n | 2 ) ⁢ u n ⁢ ( u n - u ) ¯ | ⁢ 𝑑 x → 0 as ⁢ n → ∞

and

∫ Λ ϵ ∩ B 2 ⁢ ρ ⁢ ( 0 ) | g ⁢ ( ϵ ⁢ x , | u n | 2 ) ⁢ u n ⁢ ( u n - u ) ¯ | ⁢ 𝑑 x → 0 as ⁢ n → ∞ .

Thus, (3.5) holds. Moreover, by (3.4), it follows that

0 ≤ lim sup n ⁡ ∫ B R P n ⁢ ( x ) ⁢ 𝑑 x ≤ lim sup n ⁡ ( | J n , ρ 1 | + | J n , ρ 2 | + | J n , ρ 3 | + | J n , ρ 4 | ) = 0 .

Then

lim sup n ⁡ ∫ B R P n ⁢ ( x ) ⁢ 𝑑 x = 0 .

Thus, (3.3) holds. Finally, from (3.2) and (3.3), we have

∥ u ∥ ϵ 2 ≤ lim inf n ⁡ ∥ u n ∥ ϵ 2

≤ lim sup n ⁡ ∥ u n ∥ ϵ 2

≤ lim sup n ⁡ { ∫ B R ⁢ ( 0 ) ( a ⁢ | ∇ A ϵ ⁡ u n | 2 + V ϵ ⁢ ( x ) ⁢ | u n | 2 ) ⁢ 𝑑 x + ∫ B R c ⁢ ( 0 ) ( a ⁢ | ∇ A ϵ ⁡ u n | 2 + V ϵ ⁢ ( x ) ⁢ | u n | 2 ) ⁢ 𝑑 x }

≤ ∫ B R ⁢ ( 0 ) ( a ⁢ | ∇ A ϵ ⁡ u | 2 + V ϵ ⁢ ( x ) ⁢ | u | 2 ) ⁢ 𝑑 x + ζ .

Passing to the limit as ζ→0, we have R→∞, which implies that

∥ u ∥ ϵ 2 ≤ lim inf n ⁡ ∥ u n ∥ ϵ 2 ≤ lim sup n ⁡ ∥ u n ∥ ϵ 2 ≤ ∥ u ∥ ϵ 2 .

Then un→u in Hϵ, and we complete the proof of this theorem. ∎

Since f is only assumed to be continuous, the following result is required for the multiplicity result in the next section.

Corollary 3.6.

The functional Ψϵ satisfies the (PS)c condition on Sϵ+ at any level c>0.

Proof.

Let {un}⊂Sϵ+ be a (PS)c sequence for Ψϵ. Then Ψϵ⁢(un)→c and ∥Ψϵ′⁢(un)∥*→0, where ∥⋅∥* is the norm in the dual space (Tun⁢Sϵ+)*. By Lemma 3.3 (B3), we know that {mϵ⁢(un)} is a (PS)c sequence for Jϵ in Hϵ. From Lemma 3.5, we know that there exists a u∈Sϵ+ such that, up to a subsequence, mϵ⁢(un)→mϵ⁢(u) in Hϵ. By Lemma 3.2 (A3), we obtain

u n → u in ⁢ S ϵ + ,

and the proof is complete. ∎

Proposition 3.7.

Assume that (V1)–(V2) and (f1)–(f4) hold. Then problem (3.1) has a ground state solution for any ϵ>0.

Proof.

From Lemma 3.1 and Lemma 3.5, we can obtain the existence of a ground state u∈Hϵ for problem (3.1). ∎

4 Multiple Solutions for the Modified Problem

4.1 The Autonomous Problem

For our scope, we also need to study the following limit problem:

(4.1) - ( a + b ⁢ [ u ] 2 ) ⁢ Δ ⁢ u + V 0 ⁢ u = f ⁢ ( | u | 2 ) ⁢ u , u : ℝ 3 → ℝ ,

whose associated C1-functional, defined on H1⁢(ℝ3,ℝ), is

I 0 ⁢ ( u ) := 1 2 ⁢ ∫ ℝ 3 ( a ⁢ | ∇ ⁡ u | 2 + V 0 ⁢ u 2 ) ⁢ 𝑑 x + b 4 ⁢ ( ∫ ℝ 3 | ∇ ⁡ u | 2 ⁢ 𝑑 x ) 2 - 1 2 ⁢ ∫ ℝ 3 F ⁢ ( u 2 ) ⁢ 𝑑 x .

Let

𝒩 0 := { u ∈ H 1 ⁢ ( ℝ 3 , ℝ ) ∖ { 0 } : I 0 ′ ⁢ ( u ) ⁢ [ u ] = 0 }

and

c V 0 := inf u ∈ 𝒩 0 ⁡ I 0 ⁢ ( u ) .

Let S0 be the unit sphere of H0:=H1⁢(ℝ3,ℝ) and let it be a complete and smooth manifold of codimension 1. Therefore, H0=Tu⁢S0⁢⊕ℝ⁢u for each u∈Tu⁢S0, where Tu⁢S0={v∈H0:〈u,v〉0=0}.

Lemma 4.1.

Let V0 be given in (V1) and suppose that (f1)–(f4) are satisfied. Then the following properties hold:

For any u ∈ H 0 ∖ { 0 } , let g u : ℝ + → ℝ be given by g u ⁢ ( t ) = I 0 ⁢ ( t ⁢ u ) . Then there exists a unique t u > 0 such that g u ′ ⁢ ( t ) > 0 in ( 0 , t u ) and g u ′ ⁢ ( t ) < 0 in ( t u , ∞ ) .
There is a τ > 0 independent of u such that t u > τ for all u ∈ S 0 . Moreover, for each compact 𝒲 ⊂ S 0 there exists a t u such that t u ≤ C 𝒲 for all u ∈ 𝒲 .
The map m ^ : H 0 ∖ { 0 } → 𝒩 0 given by m ^ ⁢ ( u ) = t u ⁢ u is continuous, and m 0 = m ^ 0 | S 0 is a homeomorphism between S 0 and 𝒩 0 . Moreover, m - 1 ⁢ ( u ) = u ∥ u ∥ 0 .
If there is a sequence { u n } ⊂ S 0 such that dist ⁡ ( u n , ∂ ⁡ S 0 ) → 0 , then ∥ m ⁢ ( u n ) ∥ 0 → ∞ and I 0 ⁢ ( m ⁢ ( u ) ) → ∞ as n → ∞ .

Lemma 4.2.

Let V0 be given in (V1) and suppose that (f1)–(f4) are satisfied. Then the following assertions hold:

Ψ ^ 0 ∈ C 1 ⁢ ( H 0 ∖ { 0 } , ℝ ) and

Ψ ^ 0 ′ ⁢ ( u ) ⁢ v = ∥ m ^ ⁢ ( u ) ∥ 0 ∥ u ∥ 0 ⁢ I 0 ′ ⁢ ( m ^ ⁢ ( u ) ) ⁢ [ v ] for all ⁢ u ∈ H 0 ∖ { 0 } ⁢ and all ⁢ v ∈ H 0 .
Ψ 0 ∈ C 1 ⁢ ( S 0 , ℝ ) and

Ψ 0 ′ ⁢ ( u ) ⁢ v = ∥ m ⁢ ( u ) ∥ 0 ⁢ I 0 ′ ⁢ ( m ^ ⁢ ( u ) ) ⁢ [ v ] for all ⁢ v ∈ T u ⁢ S 0 .
If { u n } is a ( PS ) c sequence of Ψ 0 , then { m ⁢ ( u n ) } is a ( PS ) c sequence of I 0 . If { u n } ⊂ 𝒩 0 is a bounded ( PS ) c sequence of I 0 , then { m - 1 ⁢ ( u n ) } is a ( PS ) c sequence of Ψ 0 .
We have that u is a critical point of Ψ 0 if and only if m ⁢ ( u ) is a critical point of I 0 . Moreover, the corresponding critical values coincide and

inf S 0 ⁡ Ψ 0 = inf 𝒩 0 ⁡ I 0 .

Similarly to the previous argument, we have the following variational characterization of the infimum of I0 over 𝒩0:

c V 0 = inf u ∈ 𝒩 0 ⁡ I 0 ⁢ ( u ) = inf u ∈ H 0 ∖ { 0 } ⁡ sup t > 0 ⁡ I 0 ⁢ ( t ⁢ u ) = inf u ∈ S 0 ⁡ sup t > 0 ⁡ I 0 ⁢ ( t ⁢ u ) .

The next result is useful in later arguments.

Lemma 4.3.

Let {un}⊂H0 be a (PS)c sequence for I0 such that un⇀0. Then one of the following alternatives occurs:

u n → 0 in H 0 as n → + ∞ .
There are a sequence { y n } ⊂ ℝ 3 and constants R, β>0 such that

lim inf n ⁡ ∫ B R ⁢ ( y n ) | u n | 2 ⁢ 𝑑 x ≥ β .

Proof.

Assume that (ii) does not hold. Then, for every R>0, we have

lim n ⁡ sup y ∈ ℝ 3 ⁡ ∫ B R ⁢ ( y ) | u n | 2 ⁢ 𝑑 x = 0 .

Since {un} is bounded in H0, by the Lions lemma it follows that

u n → 0 in ⁢ L r ⁢ ( ℝ 3 ) , 2 < r < 6 .

From the subcritical growth of f, we have

∫ ℝ 3 F ⁢ ( u n 2 ) ⁢ 𝑑 x = o n ⁢ ( 1 ) = ∫ ℝ 3 f ⁢ ( u n 2 ) ⁢ u n 2 ⁢ 𝑑 x .

Moreover, from I0′⁢(un)⁢[un]→0, it follows that

∫ ℝ 3 ( a ⁢ | ∇ ⁡ u n | 2 + V 0 ⁢ u n 2 ) ⁢ 𝑑 x + b ⁢ ( ∫ ℝ 3 | ∇ ⁡ u n | 2 ⁢ 𝑑 x ) 2 = ∫ ℝ 3 f ⁢ ( u n 2 ) ⁢ u n 2 ⁢ 𝑑 x + o n ⁢ ( 1 ) = o n ⁢ ( 1 ) .

Thus (i) holds. ∎

Remark 4.4.

From Lemma 4.3 we see that if u is the weak limit of a (PS)cV0 sequence {un} of the functional I0, then we have u≠0. Otherwise, we have that un⇀0 and if un↛0, from Lemma 4.3 it follows that there are a sequence {yn}⊂ℝ3 and constants R, β>0 such that

lim inf n ⁡ ∫ B R ⁢ ( y n ) | u n | 2 ⁢ 𝑑 x ≥ β > 0 .

Then set vn⁢(x)=un⁢(x+zn). It is easy to see that {vn} is also a (PS)cV0 sequence for the functional I0, it is bounded, and there exists v∈H0 such that vn⇀v in H0 with v≠0.

Lemma 4.5.

Assume that V0>0 and f satisfies (f1)–(f4). Then problem (4.1) has a positive ground state solution.

Proof.

First of all, it is easy to show that cV0>0. Moreover, if u0∈𝒩0 satisfies I0⁢(u0)=cV0, then m-1⁢(u0)∈S0 is a minimizer of Ψ0, so that u0 is a critical point of I0 by Lemma 4.2. Now, we show that there exists a minimizer u∈𝒩0 of I0|𝒩0. Since infS0⁡Ψ0=inf𝒩0⁡I0=cV0 and S0 is a C1 manifold, by Ekeland’s variational principle, there exists a sequence ωn⊂S0 with Ψ0⁢(ωn)→cV0 and Ψ0′⁢(ωn)→0 as n→∞. Put un=m⁢(ωn)∈𝒩0 for n∈N. Then I0⁢(un)→cV0 and I0′⁢(un)→0 as n→∞ by Lemma 4.2 (b3). Similar to the proof of Lemma 3.4, it is easy to know that {un} is bounded in H0. Thus, we have un⇀u in H0, un→u in Llocr⁢(ℝ3),1≤r<6 and un→u a.e. in ℝ3, and thus I0′⁢(u)=0. From Remark 4.4, we know that u≠0. Moreover,

c V 0 ≤ I 0 ⁢ ( u ) = I 0 ⁢ ( u ) - 1 θ ⁢ I 0 ′ ⁢ ( u ) ⁢ [ u ]

= ( 1 2 - 1 θ ) ⁢ ∥ u ∥ 0 2 + ( 1 4 - 1 θ ) ⁢ b ⁢ ( ∫ ℝ 3 | ∇ ⁡ u | 2 ⁢ 𝑑 x ) 2 + ∫ ℝ 3 ( 1 θ ⁢ f ⁢ ( u 2 ) ⁢ u 2 - 1 2 ⁢ F ⁢ ( u 2 ) ) ⁢ 𝑑 x

≤ lim inf n ⁡ { ( 1 2 - 1 θ ) ⁢ ∥ u n ∥ 0 2 + ( 1 4 - 1 θ ) ⁢ b ⁢ ( ∫ ℝ 3 | ∇ ⁡ u n | 2 ⁢ 𝑑 x ) 2 + ∫ ℝ 3 ( 1 θ ⁢ f ⁢ ( u n ) ⁢ u n 2 - 1 2 ⁢ F ⁢ ( u n 2 ) ) ⁢ 𝑑 x }

= lim inf n ⁡ { I 0 ⁢ ( u n ) - 1 θ ⁢ I 0 ′ ⁢ ( u n ) ⁢ [ u n ] }

= c V 0 ,

Thus, u is a ground state solution. From the assumption on f, we have u≥0, and thus u⁢(x)>0 for all x∈ℝN. The proof is complete. ∎

Arguing as in [5, Proposition 4], there exists a positive radial ground state solution of problem (4.1), which implies that this solution decays exponentially at infinity with its gradient; moreover, this ground state solution is of class C2⁢(ℝ3,ℝ)∩L∞⁢(ℝ3,ℝ).

Lemma 4.6.

Let (un)⊂N0 be such that I0⁢(un)→cV0. Then (un) has a convergent subsequence in H0.

Proof.

Since (un)⊂𝒩0, from Lemma 4.1 (a3), Lemma 4.2 (b4) and the definition of cV0, we have

v n = m - 1 ⁢ ( u n ) = u n ∥ u n ∥ 0 ∈ S 0 for all ⁢ n ∈ N ,

and

Ψ 0 ⁢ ( v n ) = I 0 ⁢ ( u n ) → c V 0 = inf u ∈ S 0 ⁡ Ψ 0 ⁢ ( u ) .

Since S0 is a complete C1 manifold, by the Ekeland’s variational principle, there exists a sequence {v~n}⊂S0 such that {v~n} is a (PS)cV0 sequence for Ψ0 on S0 and

∥ v ~ n - v n ∥ 0 = o n ⁢ ( 1 ) .

Similar to Lemma 4.5, we may obtain the conclusion of this lemma. ∎

4.2 The Technical Results

In this subsection, we prove a multiplicity result for the modified problem (3.1) using the Ljusternik–Schnirelmann category theory. In order to get it, we first provide some useful preliminaries.

Let δ>0 be such that Mδ⊂Λ, let ω∈H1⁢(ℝ3,ℝ) be a positive ground state solution of the limit problem (4.1), and let η∈C∞⁢(ℝ+,[0,1]) be a nonincreasing cut-off function defined in [0,+∞) such that η⁢(t)=1 if 0≤t≤δ2 and η⁢(t)=0 if t≥δ.

For any y∈M, let us introduce the function

Ψ ϵ , y ⁢ ( x ) := η ⁢ ( | ϵ ⁢ x - y | ) ⁢ ω ⁢ ( ϵ ⁢ x - y ϵ ) ⁢ exp ⁡ ( i ⁢ τ y ⁢ ( ϵ ⁢ x - y ϵ ) ) ,

where

τ y ⁢ ( x ) := ∑ i = 1 3 A i ⁢ ( y ) ⁢ x i .

Let tϵ>0 be the unique positive number such that

max t ≥ 0 ⁡ J ϵ ⁢ ( t ⁢ Ψ ϵ , y ) = J ϵ ⁢ ( t ϵ ⁢ Ψ ϵ , y ) .

Note that tϵ⁢Ψϵ,y∈𝒩ϵ.

Let us define Φϵ:M→𝒩ϵ by

Φ ϵ ⁢ ( y ) := t ϵ ⁢ Ψ ϵ , y .

By construction, Φϵ⁢(y) has compact support for any y∈M. Moreover, the energy of the above functions has the following behavior as ϵ→0+.

Lemma 4.7.

The limit

lim ϵ → 0 + ⁡ J ϵ ⁢ ( Φ ϵ ⁢ ( y ) ) = c V 0

holds uniformly in y∈M.

Proof.

Assume by contradiction that the statement is false. Then there exist δ0>0, (yn)⊂M and ϵn→0+ satisfying

| J ϵ n ⁢ ( Φ ϵ n ⁢ ( y n ) ) - c V 0 | ≥ δ 0 .

For simplicity, we write Φn, Ψn and tn for Φϵn⁢(yn), Ψϵn,yn and tϵn, respectively.

By the Lebesgue dominated convergence theorem, we have that

(4.2) ∥ Ψ n ∥ ϵ n 2 → ∫ ℝ 3 ( | ∇ ⁡ ω | 2 + V 0 ⁢ ω 2 ) ⁢ 𝑑 x ⁢ as ⁢ n → + ∞ ,

(4.3) [ Ψ n ] A ϵ n 4 → [ ω ] 4 ⁢ as ⁢ n → + ∞ .

Since Jϵn′⁢(tn⁢Ψn)⁢(tn⁢Ψn)=0, by the change of variables z=(ϵn⁢x-yn)/ϵn, observe that, if z∈Bδ/ϵn⁢(0), then ϵn⁢z+yn∈Bδ⁢(yn)⊂Mδ⊂Λ. We have

∥ Ψ n ∥ ϵ n 2 + t n 2 ⁢ b ⁢ [ Ψ n ] A ϵ n 4 = ∫ ℝ 3 g ⁢ ( ϵ n ⁢ z + y n , t n 2 ⁢ η 2 ⁢ ( | ϵ n ⁢ z | ) ⁢ ω 2 ⁢ ( z ) ) ⁢ η 2 ⁢ ( | ϵ n ⁢ z | ) ⁢ ω 2 ⁢ ( z ) ⁢ 𝑑 z

= ∫ ℝ 3 f ⁢ ( t n 2 ⁢ η 2 ⁢ ( | ϵ n ⁢ z | ) ⁢ ω 2 ⁢ ( z ) ) ⁢ η 2 ⁢ ( | ϵ n ⁢ z | ) ⁢ ω 2 ⁢ ( z ) ⁢ 𝑑 z

≥ ∫ B δ / ( 2 ⁢ ϵ n ) ⁢ ( 0 ) f ⁢ ( t n 2 ⁢ ω 2 ⁢ ( z ) ) ⁢ ω 2 ⁢ ( z ) ⁢ 𝑑 z

≥ ∫ B δ / 2 ⁢ ( 0 ) f ⁢ ( t n 2 ⁢ ω 2 ⁢ ( z ) ) ⁢ ω 2 ⁢ ( z ) ⁢ 𝑑 z

≥ f ⁢ ( t n 2 ⁢ γ 2 ) ⁢ ∫ B δ / 2 ⁢ ( 0 ) ω 4 ⁢ ( z ) ⁢ 𝑑 z

for all n large enough and where γ=min⁡{ω⁢(z):|z|≤δ2}. Moreover, we have

t n - 2 ⁢ ∥ Ψ n ∥ ϵ n 2 + b ⁢ [ Ψ n ] A ϵ n 4 ≥ f ⁢ ( t n 2 ⁢ γ 2 ) t n 2 ⁢ γ 2 ⁢ γ 2 ⁢ ∫ B δ / 2 ⁢ ( 0 ) ω 4 ⁢ ( z ) ⁢ 𝑑 z .

If tn→+∞, by (f4) we derive a contradiction.

Therefore, up to a subsequence, we may assume that tn→t0≥0. If tn→0, using the fact that f is increasing and using the Lebesgue dominated convergence theorem, we obtain that

∥ Ψ n ∥ ϵ n 2 + t n 2 ⁢ b ⁢ [ Ψ n ] A ϵ n 4 = ∫ ℝ 3 f ⁢ ( t n 2 ⁢ η 2 ⁢ ( | ϵ n ⁢ z | ) ⁢ ω 2 ⁢ ( z ) ) ⁢ η 2 ⁢ ( | ϵ n ⁢ z | ) ⁢ ω 2 ⁢ ( z ) ⁢ 𝑑 z → 0 as ⁢ n → + ∞ ,

which contradicts (4.2). Thus, from (4.2) and (4.3), we have t0>0 and

∫ ℝ 3 ( | ∇ ⁡ ω | 2 + V 0 ⁢ ω 2 ) ⁢ 𝑑 x + t 0 2 ⁢ b ⁢ [ ω ] 4 = ∫ ℝ 3 f ⁢ ( t 0 ⁢ ω 2 ) ⁢ ω 2 ⁢ 𝑑 x ,

so that t0⁢ω∈𝒩V0. Since ω∈𝒩V0, we obtain that t0=1 and so, using the Lebesgue dominated convergence theorem, we get

lim n ⁡ ∫ ℝ 3 F ⁢ ( | t n ⁢ Ψ n | 2 ) ⁢ 𝑑 x = ∫ ℝ 3 F ⁢ ( ω 2 ) ⁢ 𝑑 x .

Hence

lim n ⁡ J ϵ n ⁢ ( Φ ϵ n ⁢ ( y n ) ) = I 0 ⁢ ( ω ) = c V 0 ,

which is a contradiction and the proof is complete. ∎

Now we define the barycenter map.

Let ρ>0 be such that Mδ⊂Bρ and consider Υ:ℝ3→ℝ3 defined by setting

Υ ⁢ ( x ) := { x if ⁢ | x | < ρ , ρ ⁢ x | x | if ⁢ | x | ≥ ρ .

The barycenter map βϵ:𝒩ϵ→ℝ3 is defined by

β ϵ ⁢ ( u ) := 1 ∥ u ∥ 4 4 ⁢ ∫ ℝ 3 Υ ⁢ ( ϵ ⁢ x ) ⁢ | u ⁢ ( x ) | 4 ⁢ 𝑑 x .

We have the following lemma.

Lemma 4.8.

The limit

lim ϵ → 0 + ⁡ β ϵ ⁢ ( Φ ϵ ⁢ ( y ) ) = y

holds uniformly in y∈M.

Proof.

Assume by contradiction that there exist κ>0, (yn)⊂M and ϵn→0 such that

(4.4) | β ϵ n ⁢ ( Φ ϵ n ⁢ ( y n ) ) - y n | ≥ κ .

Using the change of variable z=(ϵn⁢x-yn)/ϵn, we can see that

β ϵ n ⁢ ( Φ ϵ n ⁢ ( y n ) ) = y n + ∫ ℝ 3 ( Υ ⁢ ( ϵ n ⁢ z + y n ) - y n ) ⁢ η 4 ⁢ ( | ϵ n ⁢ z | ) ⁢ ω 4 ⁢ ( z ) ⁢ 𝑑 z ∫ ℝ 3 η 4 ⁢ ( | ϵ n ⁢ z | ) ⁢ ω 4 ⁢ ( z ) ⁢ 𝑑 z .

Taking into account (yn)⊂M⊂Mδ⊂Bρ and the Lebesgue dominated convergence theorem, we can obtain that

| β ϵ n ⁢ ( Φ ϵ n ⁢ ( y n ) ) - y n | = o n ⁢ ( 1 ) ,

which contradicts (4.4). ∎

Now, we prove the following useful compactness result.

Proposition 4.9.

Let ϵn→0+ and (un)⊂Nϵn be such that Jϵn⁢(un)→cV0. Then there exists (y~n)⊂R3 such that the sequence (|vn|)⊂H1⁢(R3,R), where vn⁢(x):=un⁢(x+y~n), has a convergent subsequence in H1⁢(R3,R). Moreover, up to a subsequence, yn:=ϵn⁢y~n→y∈M as n→+∞.

Proof.

Since Jϵn′⁢(un)⁢[un]=0 and Jϵn⁢(un)→cV0, arguing as in Lemma 3.4, we can prove that there exists C>0 such that ∥un∥ϵn≤C for all n∈ℕ.

Arguing as in the proof of Lemma 3.2 and recalling that cV0>0, we have that there exist a sequence {y~n}⊂ℝ3 and constants R, β>0 such that

(4.5) lim inf n ⁡ ∫ B R ⁢ ( y ~ n ) | u n | 2 ⁢ 𝑑 x ≥ β .

Now, let us consider the sequence {|vn|}⊂H1⁢(ℝ3,ℝ), where vn⁢(x):=un⁢(x+y~n). By the diamagnetic inequality (2.1), we get that {|vn|} is bounded in H1⁢(ℝ3,ℝ). Using (4.5), we may assume that |vn|⇀v in H1⁢(ℝ3,ℝ) for some v≠0.

Let tn>0 be such that v~n:=tn⁢|vn|∈𝒩V0, and set yn:=ϵn⁢y~n.

By the diamagnetic inequality (2.1), we have

c V 0 ≤ I 0 ⁢ ( v ~ n ) ≤ max t ≥ 0 ⁡ J ϵ n ⁢ ( t ⁢ u n ) = J ϵ n ⁢ ( u n ) = c V 0 + o n ⁢ ( 1 ) ,

which yields I0⁢(v~n)→cV0 as n→+∞.

Since the sequences {|vn|} and {v~n} are bounded in H1⁢(ℝ3,ℝ) and |vn|↛0 in H1⁢(ℝ3,ℝ), we have that (tn) is also bounded and so, up to a subsequence, we may assume that tn→t0≥0.

We claim that t0>0. Indeed, if t0=0, then, since (|vn|) is bounded, we have v~n→0 in H1⁢(ℝ3,ℝ), that is, I0⁢(v~n)→0, which contradicts cV0>0.

Thus, up to a subsequence, we may assume that v~n⇀v~:=t0⁢v≠0 in H1⁢(ℝ3,ℝ), and, by Lemma 4.6, we can deduce that v~n→v~ in H1⁢(ℝ3,ℝ), which gives |vn|→v in H1⁢(ℝ3,ℝ).

Now we show the final part, namely that {yn} has a subsequence such that yn→y∈M. Assume by contradiction that {yn} is not bounded and so, up to a subsequence, |yn|→+∞ as n→+∞. Choose R>0 such that Λ⊂BR⁢(0). Then, for n large enough, we have |yn|>2⁢R, and, for any x∈BR/ϵn⁢(0),

| ϵ n ⁢ x + y n | ≥ | y n | - ϵ n ⁢ | x | > R .

Since un∈𝒩ϵn, using (V1) and the diamagnetic inequality (2.1), we get that

∫ ℝ 3 ( a ⁢ | ∇ ⁡ | v n | | 2 + V 0 ⁢ | v n | 2 ) ⁢ 𝑑 x ≤ ∫ ℝ 3 g ⁢ ( ϵ n ⁢ x + y n , | v n | 2 ) ⁢ | v n | 2 ⁢ 𝑑 x

(4.6) ≤ ∫ B R / ϵ n ⁢ ( 0 ) f ~ ⁢ ( | v n | 2 ) ⁢ | v n | 2 ⁢ 𝑑 x + ∫ B R / ϵ n c ⁢ ( 0 ) f ⁢ ( | v n | 2 ) ⁢ | v n | 2 ⁢ 𝑑 x .

Since |vn|→v in H1⁢(ℝ3,ℝ) and f~⁢(t)≤V0/K, we can see that (4.6) yields

min ⁡ { 1 , V 0 ⁢ ( 1 - 1 K ) } ⁢ ∫ ℝ 3 ( a ⁢ | ∇ ⁡ | v n | | 2 + V 0 ⁢ | v n | 2 ) ⁢ 𝑑 x = o n ⁢ ( 1 ) ,

that is, |vn|→0 in H1⁢(ℝ3,ℝ), which contradicts to v≢0.

Therefore, we may assume that yn→y0∈ℝ3. Assume by contradiction that y0∉Λ¯. Then there exists r>0 such that for every n large enough we have that |yn-y0|<r and B2⁢r⁢(y0)⊂Λ¯c. Then, if x∈Br/ϵn⁢(0), we have that |ϵn⁢x+yn-y0|<2⁢r so that ϵn⁢x+yn∈Λ¯c and so, arguing as before, we reach a contradiction. Thus, y0∈Λ¯.

To prove that V⁢(y0)=V0, we suppose by contradiction that V⁢(y0)>V0. Using Fatou’s lemma, the change of variable z=x+y~n and maxt≥0⁡Jϵn⁢(t⁢un)=Jϵn⁢(un), we obtain

c V 0 = I 0 ⁢ ( v ~ ) < 1 2 ⁢ ∫ ℝ 3 ( a ⁢ | ∇ ⁡ v ~ | 2 + V ⁢ ( y 0 ) ⁢ | v ~ | 2 ) ⁢ 𝑑 x + b 4 ⁢ ( ∫ ℝ 3 | ∇ ⁡ v ~ | 2 ⁢ 𝑑 x ) 2 - 1 2 ⁢ ∫ ℝ 3 F ⁢ ( | v ~ | 2 ) ⁢ 𝑑 x

≤ lim inf n ⁡ ( 1 2 ⁢ ∫ ℝ 3 ( a ⁢ | ∇ ⁡ v ~ n | 2 + V ⁢ ( ϵ n ⁢ x + y n ) ⁢ | v ~ n | 2 ) ⁢ 𝑑 x + b 4 ⁢ ( ∫ ℝ 3 | ∇ ⁡ v ~ n | 2 ⁢ 𝑑 x ) 2 - 1 2 ⁢ ∫ ℝ 3 F ⁢ ( | v ~ n | 2 ) ⁢ 𝑑 x )

= lim inf n ⁡ ( t n 2 2 ⁢ ∫ ℝ 3 ( a ⁢ | ∇ ⁡ | u n | | 2 + V ⁢ ( ϵ n ⁢ z ) ⁢ | u n | 2 ) ⁢ 𝑑 x + t n 4 ⁢ b 4 ⁢ ( ∫ ℝ 3 | ∇ ⁡ u n | 2 ⁢ 𝑑 x ) 2 - 1 2 ⁢ ∫ ℝ 3 F ⁢ ( | t n ⁢ u n | 2 ) ⁢ 𝑑 x )

≤ lim inf n ⁡ J ϵ n ⁢ ( t n ⁢ u n )

≤ lim inf n ⁡ J ϵ n ⁢ ( u n ) = c V 0 ,

which is impossible and the proof is complete. ∎

Let now

𝒩 ~ ϵ := { u ∈ 𝒩 ϵ : J ϵ ⁢ ( u ) ≤ c V 0 + h ⁢ ( ϵ ) } ,

where h:ℝ+→ℝ+, h⁢(ϵ)→0 as ϵ→0+.

For fixed y∈M, since, by Lemma 4.7, |Jϵ⁢(Φϵ⁢(y))-cV0|→0 as ϵ→0+, we get that 𝒩~ϵ≠∅ for any ϵ>0 small enough.

We have the following relation between 𝒩~ϵ and the barycenter map.

Lemma 4.10.

We have

lim ϵ → 0 + ⁡ sup u ∈ 𝒩 ~ ϵ ⁡ dist ⁡ ( β ϵ ⁢ ( u ) , M δ ) = 0 .

Proof.

Let ϵn→0+ as n→+∞. For any n∈ℕ, there exists un∈𝒩~ϵn such that

sup u ∈ 𝒩 ~ ϵ n ⁡ inf y ∈ M δ ⁡ | β ϵ n ⁢ ( u ) - y | = inf y ∈ M δ ⁡ | β ϵ n ⁢ ( u n ) - y | + o n ⁢ ( 1 ) .

Therefore, it is enough to prove that there exists (yn)⊂Mδ such that

lim n ⁡ | β ϵ n ⁢ ( u n ) - y n | = 0 .

By the diamagnetic inequality (2.1), we can see that I0⁢(t⁢|un|)≤Jϵn⁢(t⁢un) for any t≥0. Therefore, recalling that {un}⊂𝒩~ϵn⊂𝒩ϵn, we can deduce that

c V 0 ≤ max t ≥ 0 ⁡ I 0 ⁢ ( t ⁢ | u n | ) ≤ max t ≥ 0 ⁡ J ϵ n ⁢ ( t ⁢ u n ) = J ϵ n ⁢ ( u n ) ≤ c V 0 + h ⁢ ( ϵ n ) ,

which implies that Jϵn⁢(un)→cV0 as n→+∞. Then Proposition 4.9 implies that there exists {y~n}⊂ℝ3 such that yn=ϵn⁢y~n∈Mδ for n large enough.

Thus, making the change of variable z=x-y~n, we get

β ϵ n ⁢ ( u n ) = y n + ∫ ℝ 3 ( Υ ⁢ ( ϵ n ⁢ z + y n ) - y n ) ⁢ | u n ⁢ ( z + y ~ n ) | 4 ⁢ 𝑑 z ∫ ℝ 3 | u n ⁢ ( z + y ~ n ) | 4 ⁢ 𝑑 z .

Since, up to a subsequence, |un|(⋅+y~n) converges strongly in H1⁢(ℝ3,ℝ) and ϵn⁢z+yn→y∈M for any z∈ℝ3, we conclude the proof. ∎

4.3 Multiplicity of Solutions for Problem (3.1)

Finally, we present a relation between the topology of M and the number of solutions of the modified problem (3.1).

Theorem 4.11.

For any δ>0 such that Mδ⊂Λ, there exists ϵ~δ>0 such that, for any ϵ∈(0,ϵ~δ), problem (3.1) has at least catMδ⁡(M) nontrivial solutions.

Proof.

For any ϵ>0, we define the function πϵ:M→Sϵ+ by

π ϵ ⁢ ( y ) = m ϵ - 1 ⁢ ( Φ ϵ ⁢ ( y ) ) for all ⁢ y ∈ M .

By Lemma 4.7 and Lemma 3.3 (B4), we obtain

lim ϵ → 0 ⁡ Ψ ϵ ⁢ ( π ϵ ⁢ ( y ) ) = lim ϵ → 0 ⁡ J ϵ ⁢ ( Φ ϵ ⁢ ( y ) ) = c V 0 uniformly in ⁢ y ∈ M .

Hence, there is a number ϵ^>0 such that the set S~ϵ+:={u∈Sϵ+:Ψϵ⁢(u)≤cV0+h⁢(ϵ)} is nonempty for all ϵ∈(0,ϵ^) since πϵ⁢(M)⊂S~ϵ+. Here h is given in the definition of 𝒩~ϵ.

Given δ>0, by Lemma 4.7, Lemma 3.2 (A3), Lemma 4.8, and Lemma 4.10, we can find ϵ~δ>0 such that for any ϵ∈(0,ϵ~δ) the diagram

M → Φ ϵ Φ ϵ ⁢ ( M ) → m ϵ - 1 π ϵ ⁢ ( M ) → m ϵ Φ ϵ ⁢ ( M ) → β ϵ M δ

is well defined and continuous. From Lemma 4.8, we can choose a function Θ⁢(ϵ,z) with |Θ⁢(ϵ,z)|<δ2 uniformly in z∈M for all ϵ∈(0,ϵ^) such that βϵ⁢(Φϵ⁢(z))=z+Θ⁢(ϵ,z) for all z∈M. Define

H ⁢ ( t , z ) = z + ( 1 - t ) ⁢ Θ ⁢ ( ϵ , z ) .

Then H:[0,1]×M→Mδ is continuous. Clearly, H⁢(0,z)=βϵ⁢(Φϵ⁢(z)) and H⁢(1,z)=z for all z∈M. That is, H⁢(t,z) is a homotopy between βϵ∘Φϵ=(βϵ∘mϵ)∘πϵ and the embedding ι:M→Mδ. Thus, this fact implies that

(4.7) cat π ϵ ⁢ ( M ) ⁡ ( π ϵ ⁢ ( M ) ) ≥ cat M δ ⁡ ( M ) .

By Corollary 3.6 and the abstract category theorem [21], Ψϵ has at least catπϵ⁢(M)⁡(πϵ⁢(M)) critical points on Sϵ+. Therefore, from Lemma 3.3 (B4) and (4.7), we have that Jϵ has at least catMδ⁡(M) critical points in 𝒩~ϵ, which implies that problem (3.1) has at least catMδ⁡(M) solutions. ∎

5 Proof of Theorem 1.1

In this section, we prove our main result. The idea is to show that the solutions uϵ obtained in Theorem 4.11 satisfy

| u ϵ ⁢ ( x ) | 2 ≤ a 0 for ⁢ x ∈ Λ ϵ c

for ϵ>0 small. Arguing as in [26], the following uniform result holds.

Lemma 5.1.

Let ϵn→0+ and let un∈N~ϵn be a solution of problem (3.1) for ϵ=ϵn. Then Jϵn⁢(un)→cV0. Moreover, there exists {y~n}⊂R3 such that, if vn⁢(x):=un⁢(x+y~n), we have that {|vn|} is bounded in L∞⁢(R3,R) and

lim | x | → + ∞ ⁡ | v n ⁢ ( x ) | = 0 uniformly in ⁢ n ∈ ℕ .

Now, we are ready to give the proof of Theorem 1.1.

Proof of Theorem 1.1.

Let δ>0 be such that Mδ⊂Λ. We want to show that there exists ϵ^δ>0 such that for any ϵ∈(0,ϵ^δ) and any solution uϵ∈𝒩~ϵ of problem (3.1), it holds

(5.1) ∥ u ϵ ∥ L ∞ ⁢ ( Λ ϵ c ) 2 ≤ a 0 .

We argue by contradiction and assume that there is a sequence ϵn→0 such that for every n there exists un∈𝒩~ϵn which satisfies Jϵn′⁢(un)=0 and

(5.2) ∥ u n ∥ L ∞ ⁢ ( Λ ϵ n c ) 2 > a 0 .

As in Lemma 5.1, we have that Jϵn⁢(un)→cV0, and therefore we can use Proposition 4.9 to obtain a sequence (y~n)⊂ℝ3 such that yn:=ϵn⁢y~n→y0 for some y0∈M. Then we can find r>0 such that Br⁢(yn)⊂Λ, and so Br/ϵn⁢(y~n)⊂Λϵn for all n large enough.

By using Lemma 5.1, there exists R>0 such that |vn|2≤a0 in BRc⁢(0) and n large enough, where vn=un(⋅+y~n). Hence |un|2≤a0 in BRc⁢(y~n) and n large enough. Moreover, if n is so large that r/ϵn>R, then

Λ ϵ n c ⊂ B r / ϵ n c ⁢ ( y ~ n ) ⊂ B R c ⁢ ( y ~ n ) ,

which gives |un|2≤a0 for any x∈Λϵnc. This contradicts (5.2) and proves the claim.

Let now ϵδ:=min⁡{ϵ^δ,ϵ~δ}, where ϵ~δ>0 is given by Theorem 4.11. Then we have catMδ⁡(M) nontrivial solutions to problem (3.1). If uϵ∈𝒩~ϵ is one of these solutions, then, by (5.1) and the definition of g, we conclude that uϵ is also a solution to problem (2.2).

Finally, we study the behavior of the maximum points of |u^ϵ|, where u^ϵ⁢(x):=uϵ⁢(xϵ) is a solution to problem (1.1) as ϵ→0+.

Take ϵn→0+ and the sequence (un) where each un is a solution of (3.1) for ϵ=ϵn. From the definition of g, there exists γ∈(0,a) such that

g ⁢ ( ϵ ⁢ x , t 2 ) ⁢ t 2 ≤ V 0 K ⁢ t 2 for all ⁢ x ∈ ℝ N , | t | ≤ γ .

Arguing as above, we can take R>0 such that, for n large enough,

(5.3) ∥ u n ∥ L ∞ ⁢ ( B R c ⁢ ( y ~ n ) ) < γ .

Up to a subsequence, we may also assume that, for n large enough,

(5.4) ∥ u n ∥ L ∞ ⁢ ( B R ⁢ ( y ~ n ) ) ≥ γ .

Indeed, if (5.4) does not hold, up to a subsequence, if necessary, we have ∥un∥∞<γ. Thus, since Jϵn′⁢(uϵn)=0, using (g5) and the diamagnetic inequality (2.1), we obtain

∫ ℝ 3 ( a ⁢ | ∇ ⁡ | u n | | 2 + V 0 ⁢ | u n | 2 ) ⁢ 𝑑 x + b ⁢ ( ∫ ℝ 3 ( | ∇ ⁡ | u n | | 2 ) ⁢ 𝑑 x ) 2 ≤ ∫ ℝ 3 g ⁢ ( ϵ n ⁢ x , | u n | 2 ) ⁢ | u n | 2 ⁢ 𝑑 x ≤ V 0 K ⁢ ∫ ℝ 3 | u n | 2 ⁢ 𝑑 x .

Since K>2, we obtain ∥un∥=0, which is a contradiction.

Taking into account (5.3) and (5.4), we can infer that the global maximum points pn of |uϵn| belong to BR⁢(y~n), that is, pn=qn+y~n for some qn∈BR. Recalling that the associated solution of problem (1.1) is u^n⁢(x)=un⁢(x/ϵn), we can see that a maximum point ηϵn of |u^n| is ηϵn=ϵn⁢y~n+ϵn⁢qn. Since qn∈BR, ϵn⁢y~n→y0 and V⁢(y0)=V0, the continuity of V allows to conclude that

lim n ⁡ V ⁢ ( η ϵ n ) = V 0 .

The proof is now complete. ∎

Communicated by Laurent Veron

Funding source: Natural Science Foundation of Shanghai

Award Identifier / Grant number: 20ZR1413900

Award Identifier / Grant number: 18ZR1409100

Funding source: Ministry of Education and Research, Romania

Award Identifier / Grant number: PCE 137 / 2021

Funding source: Javna Agencija za Raziskovalno Dejavnost RS

Award Identifier / Grant number: P1-0292

Funding statement: C. Ji was supported by Natural Science Foundation of Shanghai (20ZR1413900, 18ZR1409100). The work of V. D. Rădulescu was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDI - UEFISCDI, project number PCE 137 / 2021, within PNCDI III. Moreover, V. D. Rădulescu acknowledges the support of the Slovenian Research Agency grant P1-0292.

Acknowledgements

The authors thank the referees for several valuable comments and remarks.

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Received: 2020-12-04

Revised: 2021-03-26

Accepted: 2021-03-27

Published Online: 2021-05-18

Published in Print: 2021-08-01

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

Articles in the same Issue

https://doi.org/10.1515/ans-2021-2130

Keywords for this article

Kirchhoff Equation; Magnetic Field; Concentration; Multiple Solutions; Variational Methods

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