Abstract and Applied Analysis
Volume 2013, Issue 1 398164
Research Article
Open Access

Existence of Prescribed L2-Norm Solutions for a Class of Schrödinger-Poisson Equation

Yisheng Huang, 

Yisheng Huang

Department of Mathematics, Soochow University, Suzhou, Jiangsu 215006, China suda.edu.cn

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Zeng Liu, 

Corresponding Author

Zeng Liu

Department of Mathematics, Soochow University, Suzhou, Jiangsu 215006, China suda.edu.cn

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Yuanze Wu, 

Yuanze Wu

Cumt College of Sciences, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China cumt.edu.cn

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First published: 12 August 2013
https://doi.org/10.1155/2013/398164
Citations: 2
Academic Editor: Kanishka Perera

Abstract

By using the standard scaling arguments, we show that the infimum of the following minimization problem: −   can be achieved for p ∈ (2,3) and ρ > 0 small, where Bρ : = {u ∈ H1(ℝ3) : ∥u∥2 = ρ}. Moreover, the properties of and the associated Lagrange multiplier λρ are also given if p ∈ (2,8/3].

1. Introduction

In this paper, we consider the nonlinear Schrödinger-Poisson type equation:
()
where  λ ∈ ℝ  is a parameter,  p ∈ (2,6), and  *  denotes the convolution. Problems like (1) have attracted considerable attentions recently since a pair  (u, λ), solution of (1), corresponds to a solitary wave of the form  ψ(x, t) = e−iλtu(x)  of the evolution equation:
()
which was obtained by approximation of a special case of Hatree-Fock equation with the frequency  λ  describing a quantum mechanical system of many particles. For more mathematical and physical background of (2), we refer to [1–4] and the references therein.
In the case that the frequency  λ  is a fixed and assigned parameter, the critical points of the following functional defined in  H1(ℝ3; ℝ):
()
are the solutions of (1), where  E(u)  is obviously well defined and is a  C1  functional for each  p ∈ (2,6)  (cf. [5]). Such case has been extensively studied by using variational methods in the past decades including the existence, nonexistence, and multiplicity of solutions; see, for example, [5–12] and the references therein.
On the other hand, the physicists are often interested in the solutions with prescribed  L2-norm and unknown frequency  λ, such a solution is called a “normalized solution,” which is associated with the existence of stable standing waves. Precisely, by a “normalized solution”  (uρ, λρ)  of (1), we mean that
()
Clearly, this kind of solutions can be obtained as the constrained critical points of the  C1  functional
()
on the constraint
()
Thus, the frequency  λρ ∈ ℝ  cannot be fixed any longer and it will appear as a Lagrange multiplier associated with the critical point  uρ  on  Bρ. Among all the critical points of  I  constrained on  Bρ, we are interested in the ones with minimal energy since the corresponded standing waves are orbitally stable under the flow of (2) and can provide us some information on the dynamics of (2). Therefore, we are reduced to study the minimization problem
()
for  p ∈ (2,10/3). Here we note that, for each  ρ > 0,if  p ∈ (2,10/3), andif  p ∈ (10/3,6)  (cf. [13, Remark 1.1] or (15) below). When  p ∈ (10/3,6)  (now), by using a mountain pass argument, it was proved in [14] that  I  has a critical point constrained on  Bρ  at a strictly positive energy level for  ρ > 0  small, and this critical point is orbitally unstable.
The main difficulty of considering (7) is the lack of compactness for the (bounded) minimizing sequence  {un} ⊂ Bρ. We recall that the necessary and sufficient condition due to Lions [15, 16] in order that any minimizing sequence for (7) is relatively compact is the strong subadditivity inequality:
()

In the range  p ∈ {8/3}∪(3,10/3), by using the standard scaling arguments, Bellazzini and Siciliano in [17] proved that (8) holds for  ρ > 0  large. In the range  p ∈ (2,3), Bellazzini and Siciliano also showed in [18] that (8) holds for  ρ > 0  small, where they developed a new abstract theorem which guarantees the following condition  (MD)  for  s > 0  small:

(MD) The functionis monotone decreasing.

We remark that their abstract theorem heavily relies on the behavior ofnear zero; that is, to use the abstract theorem, one has to verify some extra conditions, such as
()
these are unnecessary if one can show (8) by using the standard scaling arguments like [17]. However, as mentioned in [18], the authors were not sure whether (8) can be proved or not by using the standard scaling arguments if  p ∈ (2,3). Therefore, the first aim of this paper is to show that (8) holds for  ρ > 0  small when  p ∈ (2,3)  by using the standard scaling arguments. To achieve this aim, we introduce a new subset  Bρ∩𝒫  of  Bρ  (see details in Section 3), then we consider the minimization problem (7) constrained on  Bρ∩𝒫  instead of  Bρ, and we use the standard scaling arguments to prove that (8) holds for  ρ > 0  small. Moreover, we can get a specific estimate on  ρ  that allows us to obtain the sign and the behavior of the Lagrange multiplier  λρ  if  p ∈ (2,8/3]; these are not considered in [18].

The other aim of this paper is to study the properties of the Lagrange multiplier  λρ  and the ratiocorresponding to the solution  (uρ, λρ)  of (1) with. It is known that  λρ  andare interpreted in physics as the frequency and the ratio between the infimum of the energy of the standing waves with fixed charge and the charge itself, respectively, and the relevance of the energy/charge ratio for the existence of standing waves in field theories has been discussed under a general framework in [19].

Our main results read as follows.

Theorem 1. All the minimizing sequences for (7) are precompact in  H1(ℝ3; ℂ)  up to translations provided that one of the following conditions holds

  • (1)

    p ∈ (2,8/3]  and, where  S  is defined by (12);

  • (2)

    p ∈ (8/3,3)  andfor some.

In particular, (1) has a solution  (uρ, λρ) ∈ H1(ℝ3; ℂ) × ℝ  such thatand. Moreover, if the above assumption (1) holds and  (uρ, λρ)  is a solution of (1) withand, then  λρ < 0,  λρ → 0  andas  ρ → 0, respectively.

Theorem 2. Let  p ∈ (2,12/5]  and let   ρ > 0. If  (uρ, λρ)  is a solution of (1) with  ∥uρ∥2 = ρ, then we have

  • (i)

    λρ < 0,  I(uρ) < 0,  λρ → 0  as  ρ → 0  and there exists a positive constant  C1, independent of  ρ, such that  λρ ∈ (−C1, 0);

  • (ii)

    there exists a positive constant  C2, independent of  ρ, such that  I(uρ)/ρ2 ∈ (−C2, 0). In particular, if, then.

Remarks. (a) We point out that parts of Theorem 1 are already contained in [18, Theorem 4.1]. In the proof of Theorem 1, within hand, we can obtain some additional information of the Lagrange multiplier  λρ  and the ratiowhen  p ∈ (2,8/3], and these are not contained in [18, Theorem 4.1]. However, we do not know whetheris optimal or not.

(b) Theorem 2(i)  shows that (1) has only the zero solution if  p ∈ (2,12/5]  and  λ ≥ 0. In the case of  p ∈ (2,3), it was shown in [5, 20] (see also [13, Remark 1.4]) that there exists  λ0 < 0  such that (1) has only the zero solution for  λ ∈ (−∞, λ0). The nonexistence results of nonzero solutions of (1) were also discussed in [13] for  p ∈ [3,10/3].

(c) As we have anticipated, the existence of minimizers foris related to the existence and stability of the standing wave solutions to (2). For the existence of stable standing wave solutions to (2), we refer to [4, 14, 17, 18, 20, 21] and the references therein.

This paper is organized as follows. In Section 2, we give some preliminaries. Section 3 is devoted to the proof of the main theorems, especially in the proof of Theorem 1, we first define a new subset of  Bρ and then analyze the properties of minimizing sequences forconstrained on the new subset, and finally, we prove that (8) holds when  p ∈ (2,8/3]  and  p ∈ (8/3,3), respectively.

2. Preliminaries

Throughout this paper, all the functions, unless otherwise stated, are complex valued, but for simplicity we will write  Lq(ℝ3),  H1(ℝ3)  and  𝒟1,2(ℝ3)  defined in the following:
  • (i)

    Lq(ℝ3)  is the usual Lebesgue space endowed with the norm, where  q ∈ [1, ∞);

  • (ii)

    H1(ℝ3)  is the usual Sobolev space endowed with the norm

    ()

  • (iii)

    𝒟1,2(ℝ3)  is the completion ofwith respect to the norm

    ()

  • (iv)

    S  is the best Sobolev imbedding constant of  𝒟1,2(ℝ3)↪L6(ℝ3)  defined as

    ()

Moreover, the letter  C  will denote a suitable positive constant, whose value may change in the same line, and the symbol  o(1)  denotes a quantity which goes to zero. We also use  O(1)  to denote a bounded quantity.
Let  ϕu(x) = |x − y|−1* | u|2, and then, for each  u ∈ H1(ℝ3),  ϕu  is the unique solution of the Poisson equation   − Δϕ = 4π | u|2 and is usually interpreted as the Coulombian potential of the electrostatic field generated by the charge density   | u|2. Evidently, see, for example [5],
()
()
For each  ρ > 0, let  u ∈ Bρ  and  ut(x) = t3/2u(tx) (t > 0), and we have, that is,  ut ∈ Bρ. Let
()
it is clear thatfor all  ρ > 0  since  fu(t) → 0  as  t → 0.
We now recall an abstract result on the constrained minimization problem
()
where  ρ > 0,  Bρ = {u ∈ H1(ℝ3) : ∥u∥2 = ρ},is assumed, and
()
for some functional  T ∈ C1(H1(ℝ3), ℝ).

Lemma 3 (see [17], [18].)Let  T ∈ C1(H1(ℝ3), ℝ). Let  ρ > 0  and  {un} ⊂ Bρ  be a minimizing sequence forweakly convergent, up to translations, to a nonzero function. Assume that (8) holds and that

()
()
()
()
Then. In particular it follows thatand.

As pointed out in [18], Lemma 3 is a variant of the concentration-compactness principle of Lions [15, 16]. Assumption (18) shows that  T  possesses the Brizis-Lieb splitting property and (19) is the homogeneity of  T. If, in addition, the condition (8) holds, then one can show that dichotomy does not occur; that is,. Furthermore, if (20) and (21) are also fulfilled, then  {un}  strongly converges toin  H1(ℝ3). Finally we recall the following results obtained in [17, 18].

Lemma 4 (see [18].)If  p ∈ (2,3), thenfor all  ρ > 0.

Lemma 5 (see [17], Lemma 3.1.)If  p ∈ (2, 10/3), then, for every  ρ > 0, the functional  I  is bounded below and coercive on  Bρ.

Remark 6. For  p ∈ (2,3), it follows from Lemmas 4 and 5 that each minimizing sequence foris bounded from below and above by two positive constants in  𝒟1,2(ℝ3)  and  H1(ℝ3), up to a subsequence, respectively.

3. Proof of the Main Theorems

Before proving our main theorems, we need some preliminary lemmas. First, we set
()
where  ut(x) = t3/2u(tx)  with  t > 0  and  Q(u)  is a functional on  H1(ℝ3)  defined as
()
It was shown in [13, Lemma 2.1] that if  uρ  is a constrained critical point of  I  on  Bρ  associated with the Lagrange multiplier  λρ, then  Q(uρ) = 0, which is nothing but a linear combination of  〈E′(uρ), uρ〉 = 0  (recall that  E(u)  is given by (3)) and the following Pohozaev identity for (1) (cf. [5, 9])
()
The following lemma shows that  Bρ∩𝒫  is well defined.

Lemma 7. Let  p ∈ (2,3)  and let   ρ > 0. For each  u ∈ Bρ  with  I(u) < 0, there exists a unique  tu > 0  such that; moreover,.

Proof. We divide the proof into two cases.

Case   1 (p ∈ (2,8/3)). Let  u ∈ Bρ, for simplicity, and we will write,and, the derivatives of  fu(t)  on  t, instead of  dfu(t)/dt,  d2fu(t)/dt2  and  d3fu(t)/dt3. From (15), we have

()
Noting that  (3p − 8)/2 ∈ (−1,0)  since  p ∈ (2,8/3), then, by (25),and; thus there exists  tu > 0  such that. If there exists another  su > 0  such that, without loss of generality, we may assume that  su > tu, and then we get
()
a contradiction. Therefore,  tu  is unique and it is clear that. Moreover,because of.

Case  2   (p ∈ [8/3,3)). By Lemma 4, we know that the set  Aρ : = {u ∈ Bρ : I(u) < 0} ≠ ∅. Let  u ∈ Aρ, iffor all  t > 0; that is,  fu(t)  is strictly increasing, then we obtain that  fu(t) < fu(1) = I(u) < 0  for all  t ∈ (0,1). However, it is easy to see that  lim t→0fu(t) = 0; this is a contradiction. On the other hand, we know that  fu(t) → ∞  as  t → ∞; hence there is a  tu > 0  such that,and

()
Next, we will show that  tu  is unique. Arguing by contradiction, suppose that there is another  su > 0  such that  fu(tu) = fu(su) = min  {fu(t) : t > 0}, without loss of generality, we may assume that  su > tu, and then we have
()
According to (28), there exists  ωu ∈ (tu, su)  such that. After a simple calculation, we get
()
()
If  p = 8/3, then, by (29),for all  t > 0, which contradicts. If  p ∈ (8/3,3), then, by (30),for all  t > 0. Noting that  ωu ∈ (tu, su), we have
()
again a contradiction. Therefore,  tu > 0  is unique.

Lemma 8. Let  p ∈ (2,3)  and  ρ > 0. For each  {un} ⊂ Bρ  such thatas  n → ∞  and  I(un) < 0  for all  n ∈ ℕ, there exists a bounded sequence  {tn} ⊂ ℝ+  such thatandas  n → ∞  withfor all  n ∈ ℕ; that is,is also a minimizing sequence forconstrained on  Bρ.

Proof. It follows from Lemma 7 that, for each  un, there exists  tn > 0  such thatand; therefore, we have

()
as  n → ∞, that is,is a minimizing sequence. Next, we will show that  {tn}  is bounded. Indeed, from Remark 6,  {un}  andare bounded from below and above by two positive constants in  𝒟1,2(ℝ3)  and  H1(ℝ3), respectively. Noting that; therefore,  {tn}  is bounded from below and above by two positive constants.

Remark 9. Thanks to the Lemma 8, we know that, and, in the following, we will consider the minimization problem (7) restricted to  Bρ∩𝒫  instead of  Bρ. By Lemmas 4 and 8, for each  ρ > 0, if  {un} ⊂ Bρ∩𝒫  satisfyingas  n → ∞, then, up to a subsequence, we may assume that  I(un) < 0. It follows from Lemma 5 that  {un}  is bounded in  H1(ℝ3); by the results of [17, 18], we may assume that  un⇀u ≠ 0  as  n → ∞  in  H1(ℝ3).

The following estimates of the elements of  Bρ∩𝒫  are crucial to proving the strong subadditivity inequality (8).

Lemma 10. Let  p ∈ (2,3)  and  ρ > 0. For each  u ∈ Bρ∩𝒫, it holds

()

Proof. Since  u ∈ Bρ∩𝒫,

()
Noting that(see (13)), by using the Hölder inequality, we get
()
which implies that
()
On the other hand, we have
()
this concludes the proof of this lemma.

Remark 11. Let  p ∈ (3,10/3). It was shown in [13, Theorem 1.1] thatif and only if, where the positive numberis defined as

()
Therefore, after a simple calculation, we can show that both of Lemmas 7 and 10 hold if  p ∈ (3,10/3)  and.

Motivated by [17], we will use the standard scaling arguments to prove that the strong subadditivity inequality (8) holds for  p ∈ (2,3). First, we consider the case of  p ∈ (2,8/3].

Lemma 12. For  p ∈ (2,8/3], let

()
Then
()

Proof. By Lemma 8 and Remark 9, for each  {un} ⊂ Bρ∩𝒫  satisfyingas  n → ∞, we may assume that, for all  n,, which implies that

()
Noting that  tun ∈ Btρ  (t > 0), we have
()
where
()
We calculate the derivative of  g(t, u)  on  t:
()
Letting  dg(t, u)/dt = 0, we see from (14) that
()
Furthermore,
()

Now we divide the value of p into two cases to discuss  dg(t, un)/dt.

Case  1 (p ∈ (2,12/5)). It follows from Lemma 10, (14), and the Hölder inequality that

()

Case  2 (p ∈ [12/5,8/3]). Again by Lemma 10, (14), and the Hölder inequality, we have

()
Let
()
Then by (47), (48), and (49), we know that, for each, there holds
()
for all  n ∈ ℕ. On the other hand, for each, it follows from (41), (44), (45), (46), and Lemma 10 that, for alland all  n ∈ ℕ,
()
This, together with the mean value theorem and (41), yields that for alland all  n ∈ ℕ,
()
whereand  C > 0  depend only on  ε,  p, and  ρ. By (42), we have
()
then
()
Clearly,(cf. (50)) is strictly increasing on  ρ, and then  is strictly decreasing on  ρ  since  p ∈ (2,3).

Let

()
For each, let  μ ∈ (0, ρ) without loss of generality, we may assume that. Choosing, then by (50) we know that.
  • (a)

    Ifρ/μ ∈ (1, h(μ)), then by (54)

    ()

  • (b)

    Ifρ/μ ∉ (1, h(μ)), then there exists  k ∈ ℕ  such that  (ρ/μ) 1/k ∈ (1, h(ρ)). Therefore

    ()

It follows from (54) that

()
Combining the above cases (a) and (b), we can show that
()
Thus the conclusion of this lemma holds.

Remark 13. For the case of  p = 8/3, it has been proved in [4, 17] that the strong subadditivity inequality (8) holds for  ρ > 0  small. By using the result of [17], we can give a specific estimate of lower bound of  ρ  such that (8) holds; that is, (8) holds for all  ρ ∈ (0, (8π) −3/4S3/2). However, if we plug  p = 8/3  into (49), then we have, which coincides with the one given in [17].

Next, we will show (8) for  p ∈ (8/3,3). We point out that the case of  p ∈ (8/3,3)  is quite different from the case of  p ∈ (2,8/3]  since the inequality (48) does not hold anymore. Inspired by [18], we will give some estimates forin Lemmas 14 and 15, and these are crucial for the proof of (8) if  p ∈ (8/3,3).

Lemma 14. Let  p ∈ (8/3,3)  and  ρ > 0  be fixed. If there exists  u ∈ Bρ∩𝒫  such thatand

()
then there exist positive constants  C3  and  C4  dependent on  p  and  ρ, such that
()

Proof. From the assumptions of the lemma, we see that

()
()
By (60), (62), and (63), we deduce that
()
Combining (62) and (64), and using Lemma 10, we also obtain
()
For each  t > 0, let  ut(x) = t4/(10−3p)u(t2(p − 2)/(10−3p)x), we have. It follows from (60), (64), and (65) that
()
Set  tρ = μ, then  μ ∈ (0, ∞)  since  t ∈ (0, ∞)  and  ρ  is a fixed positive constant. From the above inequality, we see that
()
for some positive constants  C3  and  C4  depending on  p  and  ρ.

Lemma 15. Suppose that  p ∈ (8/3,3)  andsatisfyingandfor all  k ∈ ℕ. Then there exists a positive constant  C  dependent on  p, such that

()

Proof. Following the line of the proof of Lemma 14, we arrive that

()
which, together with (14) and the Höder inequality, implies that
()
and then
()
Combining (62), (69), and (71), we have
()
and this completes the proof.

Lemma 16. If  p ∈ (8/3,3), then there exists a positive constantsuch that

()

Proof. Suppose that  ρ > 0  and  {un} ⊂ Bρ∩𝒫  satisfyingas  n → ∞. It follows from Remark 9 that, up to a subsequence,for all  n ∈ ℕ. By Lemma 5, it is easy to see that  {un}  is bounded in  H1(ℝ3). Noting that  tun ∈ Btρ, then, by (42), we have

()
where  g(t, u)  is given by (43). Obviously,
()
since  un ∈ Bρ∩𝒫  and (34) holds. Moreover,
()
for all  t > 0  and  n ∈ ℕ.

We claim that there existssuch that for eachand for each  {un} ⊂ Bρ∩𝒫  satisfyingandas  n → ∞, we have

()

Indeed, if not, we can find  {ρk}  andsuch that  ρk → 0  as  k → ∞  and for each  k ∈ ℕ,as  n → ∞, but. For  k = 1, there exists  n1 > 0  such that, and it can be deduced from Lemma 14 that

()
where  C3  and  C4  are positive constants dependent on  p  and  ρ1. On the other hand, we know that for each  k ∈ ℕ  there exists  nk > 0  such that. Then by Lemma 15, we obtain
()
where  C  is a positive constant depending only on  p. Noting that (78) holds for all  μ > 0, by (78) and (79), we deduce that
()
which is a contradiction for  k  large since  p ∈ (8/3,3)  implies
()
Thus we have shown the claim. Now for eachand for all  {un} ⊂ Bρ∩𝒫  withandas  n → ∞, using (77), we have
()
By (76), similarly as in the proofs of (45) and (51), we get that
()
Now, we can choose  ε > 0  so small that there exists a positive constant  C  dependent on  p,  ρ, and  ε, such that
()
Since, for each  n,  g(1, un) = 0, it follows that, for each  t ∈ (1, (3(1 − ε)(p − 2)/(7p − 18)) 1/(4−p)),
()
where, namely,for all  t ∈ (1, (3(1 − ε)(p − 2)/(7p − 18)) 1/(4−p)). Thus we complete the proof of this lemma by using the arguments in the proof of Lemma 12.

Lemma 17. Let  ρ > 0. Assume that  (uρ, λρ)  is a solution of (1) with.

  • (a)

    If  p ∈ (2,12/5], then  λρ < 0.

  • (b)

    If  p ∈ (12/5,8/3]  and  λρ ≥ 0, then

    ()

Proof. Since  (uρ, λρ)  is a solution of (1), it follows that

()
()
Thus, from (87) and (88), after a simple calculation, we have
()
which yields that  (a)  holds. Moreover, if  p ∈ (12/5,8/3]  and  λρ ≥ 0, then (89) implies that
()
Thus we get from (88) that
()
By using the Hölder inequality, it can be deduced from (91) and Lemma 10 that
()
and this means that
()
Thus  (b)  holds. At this point, the lemma is proved.

Proof of Theorem 1. It follows from Lemmas 12 and 16 that (8) holds. Let. From the results of [17, 18], we know that (18), (19), (20), and (21) hold. Therefore, by Lemma 3, all the minimizing sequences for (7) are precompact and then (1) has a solution  (uρ, λρ). Lemma 17 shows that, for  p ∈ (2,8/3],  λρ < 0  since, whereandare given by (49) and (86), respectively.

To complete the proof of Theorem 1, we need to show that  λρ → 0  andas  ρ → 0  provided that the assumption  (1)  of Theorem 1 holds. Indeed, since  (uρ, λρ)  is the solution of (1), it follows from (87), (88), and Lemma 10 that

()
which implies that
()
and that is,  λρ → 0  as  ρ → 0. On the other hand, we have
()
this, together with (87) and (88), gives
()
Therefore,  I(uρ) < 0  since  λρ < 0  and  p ∈ (2,8/3). Noting that, by Lemma 10 and (97), we obtain
()

Proof of Theorem 2. Suppose that  p ∈ (2,12/5]  and  (uρ, λρ)  is a solution of (1) with. Then Lemma 17 and the above proof of Theorem 1 show that  λρ < 0,  I(uρ) < 0  and  λρ → 0  as  ρ → 0. It was proved in [5, 20] (see also [13, Remark 1.4]) that there exists  λ0 < 0  such that (1) has only the zero solution when  p ∈ (2,3)  and  λ ∈ (−∞, λ0). Therefore,  λρ  must be bounded; that is,  (i)  holds. For  (ii), it is clear that (87), (88), and (96) hold; after a simple calculation, we have

()
On the other hand, sinceis the solution of the Poisson equation   − Δϕ = 4π | uρ|2, multiplying this equation by   | uρ |   and integrating, we obtain
()
It follows from (99) and (100) that
()
which implies that
()
Therefore we get
()
so that there exists  C > 0  such that  I(uρ)/ρ2 ∈ (−C, 0)  since, by  (i),  λρ < 0  is bounded.

Acknowledgments

The authors would like to thank the referees for carefully reading the paper and helpful comments, which greatly improve the presentation of the paper, in particular, the proof of Lemma 17 that was simplified. This paper is supported by Natural Science Foundation of China (11071180, 11171247) and GIP of Jiangsu Province (CXZZ12_0802).

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