Existence and Uniqueness of Multi-Bump Solutions for Nonlinear Schrödinger–Poisson Systems

1 Introduction

In this paper, we consider the following Schrödinger–Poisson system:

where p∈(4,6), ε>0 is a parameter, and K and V are nonnegative functions defined on ℝ3 that stand for the density charge and the effective potential, respectively. When the nonlinear term |u|p-2⁢u equals zero, then the wave function u denotes the stationary solution for a quantum system introduced by Benci and Fortunato [3], where system (SP) was modeled to describe the interaction of a charged particle with an electromagnetic field, while the time-independent ϕ representing electrostatic potential depends on u according to the Maxwell equations. The Schrödinger–Poisson equations are thus also called the Schrödinger–Maxwell equations. We refer to [4, 5, 32, 33] and the references cited therein for more details on physical aspects of this system. Due to these deep physics backgrounds in semiconductor theory and quantum mechanics models, such a system (SP) has been extensively studied on the existence of radial solutions, ground state solutions and multiple solutions; see, for example, [1, 2, 30, 34, 36, 40].

When K≡0, then (SP) reduces to the nonlinear Schrödinger equation of the form

In recent decades, problem (1.1) has attracted much attention on the study of the limiting behaviors of the semi-classical states, that is, the concentration of the solutions for (1.1) as ε→0. In some sense, this suggests the transition between quantum mechanics and classical mechanics. Particularly, we focus on the critical frequency case, i.e., infℝN⁡V=0, which was firstly proposed by Beyon and Wang [8, 9]. As shown in [8, 9], the new concentration phenomenon of semi-classical solutions are quite different from the situation where infℝN⁡V>0 (see [17, 19, 23]). To be precise, these solutions decaying to 0 concentrate around local or global minimum points of V as ε→0, and their decay rate is dependent on the decay rate of V to 0 around these minimum points. Afterwards, [12, 13] succeeded in obtaining the existence of multi-bump solutions for (1.1) in the flat case and infinite case introduced by [8], respectively. Their main tools are variational methods and penalization techniques. Furthermore, Beyon and Oshita [6] constructed multi-bump solutions joining solutions of different energy scales via the Lyapunov–Schmidt reduction methods provided that the solutions of limit problems are non-degenerate.

Let us go back to the Schrödinger–Poisson equations and briefly recall some known results on the existence and concentration of semi-classical solutions. In [35], Ruiz and Vaira considered the following Schrödinger–Poisson equations:

They used the Lyapunov–Schmidt reduction methods to construct multi-bump solutions concentrating around a local minimum of V with 4<p<6. The similar reduction techniques were successfully used by other authors to study single-bump solutions of (1.2) in [29] and radially symmetric solutions concentrating on the spheres in [28]. Notice that the associated limit problem of (1.2) is the classical Schrödinger equation while the one of (SP) still consists of Schrödinger–Poisson equations. Thus there has been a great deal of literature investigating the concentration of semi-classical solutions of (SP) by using variational methods; see [25] for (SP) with K≡1 and general nonlinearity f⁢(u), [26, 27] for (SP) with a suitable perturbation f and p=6, and [37] for (SP) with nonlinearity b⁢(x)⁢f⁢(u).

It is worth pointing out that the semi-classical results mentioned previously for Schrödinger–Poisson equations (1.2) or (SP) need the subcritical frequency condition that infℝ3⁡V>0. Involving the critical frequency case, however, little literature has been carried out. At least to the best of our knowledge, the only paper considering the semi-classical solutions for system (SP) is due to Zhang and Xia [39], where they assumed the following conditions:

1. V ∈ C ⁢ ( ℝ 3 ) and 0=infx∈ℝ3⁡V⁢(x)<lim inf|x|→∞⁡V⁢(x).

1. K ∈ C ⁢ ( ℝ 3 ) ∩ L ∞ ⁢ ( ℝ 3 ) and infx∈ℝ3⁡K⁢(x)=0.

1. Assume that 𝒵:={x∈ℝ3:V⁢(x)=K⁢(x)=0}≠∅ and assume a positive function W∈𝒞⁢(ℝ3) such that for every a∈𝒵 either (i)

   lim x → a ⁡ V ⁢ ( x ) | x - a | γ = ∞

   or (ii)

   lim x → a ⁡ V ⁢ ( x ) - W ⁢ ( x - a ) | x - a | γ = 0

   holds. Here, W∈𝒞⁢(ℝ3) is a positive γ-homogeneous function for some γ>0, i.e.,

   W ⁢ ( x ) > 0   for  ⁢ x ≠ 0 , and    W ⁢ ( t ⁢ x ) = t γ ⁢ W ⁢ ( x )   for all  ⁢ ( t , x ) ∈ [ 0 , ∞ ) × ℝ 3 .

   Additionally, there exists at least one point a∈𝒵 such that alternative (ii) holds, and for any such a point that satisfies (ii) there holds

   lim x → a ⁡ K ⁢ ( x ) | x - a | p - 3 p - 2 ⁢ γ = k ⁢ ( a ) ∈ [ 0 , ∞ ) .

   Such points will be collected and consist of the point set {ai:i∈ℐ}.

Then they proved that (SP) admits a positive ground state solution vε and

where a∈{ai:i∈ℐ} and w0 is a ground state solution of the following equation:

Actually, [39] also discussed the another interesting case where the potential K accumulates polynomially to zero on 𝒵 such that K⁢(x)=O⁢(|x-a|γ′) with γ′<p-3p-2⁢γ. Clearly, solutions of (SP) in [39] are of single bump.

Motivated by the above facts, in this paper we are concerned with the existence of multi-bump solutions for the Schrödinger–Poisson equations (SP) with infℝ3⁡V=infℝ3⁡K=0. To state our main results, throughout this article, we assume that there exist disjoint isolated compact sets

and Zi for i=1,…,k satisfies one of the following assumptions:

1. Z i = { a i } and there exist a small constant r>0 and a constant γ>0 such that

   V ⁢ ( x ) = | x - a i | γ + o ⁢ ( | x - a i | γ )   and   K ⁢ ( x ) = o ⁢ ( | x - a i | p - 3 p - 2 ⁢ γ ) , x ∈ B r ⁢ ( a i ) .

2. Ω i = int ⁡ Z i is a non-empty bounded domain with smooth boundary.

3. Z i = { b i } and there exist a small constant r>0 and a positive function

   a ⁢ ( x ) ∈ 𝒞 ⁢ ( ℝ 3 ∖ { 0 } )

   with a⁢(t⁢x)=t-1⁢a⁢(x) for all t>0 and x∈ℝ3∖{0} such that, for x∈Br⁢(bi),

   V ⁢ ( x ) = exp ⁡ ( - a ⁢ ( x - b i ) m - Q ⁢ ( x - b i ) )   and   K ⁢ ( x ) = exp ⁡ ( - p - 3 p - 2 ⁢ a ⁢ ( x - b i ) m - Q ⁢ ( x - b i ) ) ,

   where m is a positive constant and lim|x|→0⁡|x|m⁢Q⁢(x)=0. In addition, we also suppose that

   Ω i = { x ∈ ℝ 3 : a ⁢ ( x ) > 1 }

   is strictly star-shaped with respect to 0.

These three cases (H1)–(H3) are called finite case, flat case and infinite case, respectively, in [8]. Following [6], to investigate refined asymptotic profiles of multi-bump solutions uε, we also define the following rescaled functions for i=1,…,k:

where yi∈Zi and

Then wεi⁢(x) tends to the least energy solution wi of

with

Here, Ω0⊂ℝ3 and L∈𝒞⁢(Ω0,ℝ). To be precise, note that we choose W⁢(x)=|x|γ and k⁢(a)=0 in (H0); then (H0) coincides with (H1). It follows from (1.3) that L⁢(x)=|x|γ and Ω0=ℝ3 in (L${}_{i}$), that is,

provided that Zi fulfills (H1). For simplicity, throughout this article, we assume that yi=0 if (H2) or (H3) holds. Then problem (L${}_{i}$) corresponds to

provided that Zi fulfills (H2) or (H3), i.e., L⁢(x)=0 and Ω0=Ωi in (L${}_{i}$). Here, Ωi is introduced in (H2) or (H3).

Then we have the following result.

In the proof of Theorem 1.1, we first employ the penalization arguments like [12, 13] to construct a suit modification on the nonlinearity. Different from [12, 13], we only need one penalty functional. Then we shall show that the modified problem possesses a positive solution. Finally, it is checked that, for ε small enough, this solution also solves the original problem (SP) and admits all desired properties. We would like to point out that both [12] and [13] considered only one of three typical cases above. However, we treat all of them and prove that the heights of these bumps for uε obtained in Theorem 1.1 are of different order. In addition, note that [12, 13, 6] studied the local problems (the Schrödinger equations). Thus, the approaches in [12, 13, 6] can not be applied to this paper because of the presence of the nonlocal term u2∗14⁢π⁢|x|.

For the Schrödinger–Poisson system (SP), we used in [38] penalization arguments similar to [12, 13] to obtain the results of Theorem 1.1. However, [38] only considered two cases: flat case and infinite case, where the limit problems of (SP) are the same (see (1.6)). As discussed above, here we need one penalty functional. Moreover, condition (H3) generalizes the infinite case in [38]. So we further improve the results in [38].

Another aim of this paper is to show the uniqueness of multi-bump solutions for (SP) with critical frequency. Recently, the academic community has extensively explored the uniqueness results for other problems; see, e.g., [10, 11, 14, 15, 16, 18, 22, 24, 31] and the references therein. More precisely, to obtain the uniqueness result, [10, 22] required that the solutions of (1.1) concentrate at any non-degenerate critical points of V. Very recently, Cao, Li and Luo [11] improved these results by assuming the following conditions:

1. V is a bounded 𝒞1-function and infx∈ℝ3⁡V⁢(x)>0.

2. There exist γ>1 and a small constant δ>0 such that

   { V ⁢ ( x ) = V ⁢ ( a i ) + ∑ j = 1 N b i , j ⁢ | x j - a i , j | γ + O ⁢ ( | x - a i | γ + 1 ) , x ∈ B δ ⁢ ( a i ) , ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j = γ ⁢ b i , j ⁢ | x j - a i , j | γ - 2 ⁢ ( x j - a i , j ) + O ⁢ ( | x - a i | γ ) , x ∈ B δ ⁢ ( a i ) ,

   where x=(x1,…,xN),ai=(ai,1,…,ai,N)∈ℝN, and bi,j∈ℝ with bi,j≠0 for each i=1,…,m and j=1,…,N.

To get the uniqueness, they combined the local Pohozaev identity introduced by [18] with the decomposition technique for multi-bump solutions in [10]. Subsequently, by using the idea of [11], Li, Luo, Peng, Wang and Xiang [31] demonstrated that the perturbed Kirchhoff equations admit the unique solution concentrating at x0, the critical points of V. In addition, one can also refer to [18] for the applications of uniqueness results.

We now emphasize that the subcritical frequency condition that infx∈ℝ3⁡V⁢(x)>0 plays a fundamental role in their proof of the above unique results. To the best of our knowledge, little research has been conducted to show the uniqueness of single-bump or multi-bump solutions for the above elliptic problems with critical frequency. To state our results, we first assume that there exist disjoint isolated compact sets Z1,…,Zm⊂𝒵 satisfying (H1). Then V and K satisfy the following condition:

1. V and K are bounded 𝒞1-functions and satisfy the expansions

   ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j = γ ⁢ | x j - a i , j | γ - 2 ⁢ ( x j - a i , j ) + O ⁢ ( | x - a i | γ )   and   ∂ ⁡ K ⁢ ( x ) ∂ ⁡ x j = O ⁢ ( | x - a i | p - 3 p - 2 ⁢ γ ) , x ∈ B δ ⁢ ( a i ) ,

   where x=(x1,x2,x3) and ai=(ai,1,ai,2,ai,3) for each i=1,…,m and j=1,2,3.

Let H be the completion of 𝒞0∞⁢(ℝ3) with respect to the norm

Based on the results of Theorem 1.1, we may assume that problem (SP) has a solution uε satisfying

where xi,ε→ai as ε→0. Then we state the following result.

To prove the unique result, we will use the way of contradiction introduced in [11]. Precisely, assume that there are two different solutions uε(1) and uε(2) fulfilling (1.7). Then we take

and prove that |ξε|∞→0 as ε→0, which contradicts the definition of ξ, and so the uniqueness holds.

Compared with [11], there are some differences. Firstly, because of the presence of the Poisson term, we can not obtain a decomposition form of solutions similar to that in [11], which is the key for the estimate |ξε|∞=o⁢(1). To overcome this obstacle, by considering the rescaled function

we use the decay property of wε to simplify some calculations in [11] and get the desired estimates. Secondly, the local Pohozaev type identity (see (4.3)) and the equation (see (4.12)) that ξε satisfies are both more sophisticated due to the Poisson term again. Thirdly, as mentioned previously, here the limit problem is (1.5) due to the critical frequency case. Fortunately, it is shown in [7] that the positive solution w for problem (1.5) is non-degenerate in the sense that, for any w¯∈H1⁢(ℝ3), the problem

has only a trivial solution w¯≡0.

2 Preliminary Results

For any open set D⊂ℝ3, we consider the Sobolev space H1⁢(D) endowed with the standard inner product (u,v)D=∫D∇⁡u⁢∇⁡v+u⁢v and norm ∥u∥D=(u,u)D1/2. If D=ℝ3, we denote ∥u∥D by ∥u∥ for simplicity. When V satisfies (V0), then H⊂H1⁢(ℝ3). Now we outline the variational framework of problem (SP) and give some preliminary lemmas. Recall that by the Lax–Milgram theorem, for every u∈H1⁢(ℝ3) and any fixed ε>0, the Poisson equation -ε2⁢Δ⁢ϕ=K⁢(x)⁢u2 has a unique positive solution ϕK,u/ε∈D1,2⁢(ℝ3) given by

Substituting ϕK,u/ε into (SP), we get an equivalent equation:

Clearly, (u,ϕK,u/ε) is a solution of system (SP) if and only if u is a solution of equation (2.1). Formally, solutions of (2.1) are critical points of the functional

It is well known (see [34, 39]) that

where the linear functional Tv is defined for every v∈H1⁢(ℝ3) on D1,2⁢(ℝ3) by

Then we give some properties, which have been proved in [40].

3 Proof of Theorem 1.1

In this section, we will give the proof of Theorem 1.1. From now on, we assume that the conditions of Theorem 1.1 hold. Without loss of generality, we assume that Zi10⁢r∩(𝒵∖Zi)=∅ for suitable r. Although the functional Iε does not satisfy the (PS) condition, the penalization techniques allow us to recover the (PS) condition for a modification of the functional Iε with a modified nonlinearity. Set u+=max⁡{u,0} and u+s=(u+)s for some constants s. Set Λ=⋃i=1mZi4⁢r and let ν>0 be such that

We define

and

where χΛ is the characteristic function of the set Λ. We readily get that

We next define

Choose μ>2⁢p2-p+32. Let M1 be a positive constant to be determined later. Then we define

where, and in the sequel, |u|2,ℝ3∖Λ2:=∫ℝ3∖Λu2. We now define the penalized functional Eε:H→ℝ by

It follows that these functionals Lε and Qε are of class 𝒞1, and so is Eε. As previously mentioned, the following lemma shows that Eε satisfies the (PS) condition.

We give the following notations used frequently. For i=1,…,k, let

Then V~εi⁢(x) and K~εi⁢(x) have the following properties.