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

Abstract

By using the standard scaling arguments, we show that the infimum of the following minimization problem: 𝐼_𝜌2=inf{(1/2)∫_ℝ3|∇𝑢|²𝑑𝑥+(1/4)∬_ℝ3(|𝑢(𝑥)|²|𝑢(𝑦)|²/|𝑥−𝑦|)𝑑𝑥𝑑𝑦 −  (1/𝑝)∫_ℝ3|𝑢|^𝑝𝑑𝑥∶𝑢∈𝐵_𝜌} can be achieved for 𝑝∈(2,3) and 𝜌>0 small, where 𝐵_𝜌∶={𝑢∈𝐻¹(ℝ³)∶‖𝑢‖₂=𝜌}. Moreover, the properties of 𝐼_𝜌2/𝜌² and the associated Lagrange multiplier 𝜆_𝜌 are also given if 𝑝∈(2,8/3].

1. Introduction

In this paper, we consider the nonlinear Schrödinger-Poisson type equation: −Δ𝑢+|𝑥|⁻¹∗|𝑢|²𝑢−|𝑢|^𝑝−2𝑢=𝜆𝑢,inℝ³, (1) where𝜆∈ℝis 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 form𝜓(𝑥,𝑡)=𝑒^{−𝑖𝜆𝑡}𝑢(𝑥)of the evolution equation: 𝑖𝜓_𝑡+Δ𝜓−|𝑥|⁻¹∗𝜓²𝜓+𝜓^𝑝−2𝜓=0,inℝ³×ℝ⁺, (2) which was obtained by approximation of a special case of Hatree-Fock equation with the frequency𝜆describing 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 frequency𝜆is a fixed and assigned parameter, the critical points of the following functional defined in𝐻¹(ℝ³;ℝ): 𝐸(𝑢)=12_ℝ³|∇𝑢|²𝜆𝑑𝑥−2_ℝ³|𝑢|²+1𝑑𝑥4_ℝ³|𝑢(𝑥)|²𝑢𝑦²1𝑥−𝑦𝑑𝑥𝑑𝑦−𝑝_ℝ³|𝑢|^𝑝𝑑𝑥, (3) are the solutions of (1), where𝐸(𝑢)is obviously well defined and is a𝐶¹functional for each𝑝∈(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 prescribed𝐿²-norm and unknown frequency𝜆, such a solution is called a “normalized solution,” which is associated with the existence of stable standing waves. Precisely, by a “normalized solution”(𝑢_𝜌,𝜆_𝜌)of (1), we mean that 𝑢_𝜌,𝜆_𝜌∈𝐻¹ℝ³;ℂ×ℝsolves(1)with𝑢_𝜌2=𝜌. (4) Clearly, this kind of solutions can be obtained as the constrained critical points of the𝐶¹functional 𝐼(𝑢)=12_ℝ³|∇𝑢|²1𝑑𝑥+4_ℝ³|𝑢(𝑥)|²𝑢𝑦²−1𝑥−𝑦𝑑𝑥𝑑𝑦𝑝_ℝ³|𝑢|^𝑝𝑑𝑥, (5) on the constraint 𝐵_𝜌=𝑢∈𝐻¹ℝ³;ℂ∶‖𝑢‖₂=𝜌. (6) Thus, the frequency𝜆_𝜌∈ℝcannot be fixed any longer and it will appear as a Lagrange multiplier associated with the critical point𝑢_𝜌on𝐵_𝜌. Among all the critical points of𝐼constrained on𝐵_𝜌, 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 𝐼_𝜌2=min_{𝑢∈𝐵𝜌}𝐼(𝑢), (7) for𝑝∈(2,10/3). Here we note that, for each𝜌>0,𝐼_𝜌2∈(−∞,0]if𝑝∈(2,10/3), and𝐼_𝜌2=−∞if𝑝∈(10/3,6)(cf. [13, Remark 1.1] or (15) below). When𝑝∈(10/3,6)(now𝐼_𝜌2=−∞), by using a mountain pass argument, it was proved in [14] that𝐼has a critical point constrained on𝐵_𝜌at a strictly positive energy level for𝜌>0small, and this critical point is orbitally unstable.

The main difficulty of considering (7) is the lack of compactness for the (bounded) minimizing sequence{𝑢_𝑛}⊂𝐵_𝜌. 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: 𝐼_𝜌2<𝐼_𝜇2+𝐼_{𝜌2−𝜇2},∀0<𝜇<𝜌. (8)

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

(MD) The function𝑠↦𝐼_𝑠2/𝑠²is monotone decreasing.

We remark that their abstract theorem heavily relies on the behavior of𝐼_𝜌2near zero; that is, to use the abstract theorem, one has to verify some extra conditions, such as 𝜌↦𝐼_𝜌2iscontinuous;lim_𝜌→0𝐼_𝜌2𝜌²=0; (9) 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 if𝑝∈(2,3). Therefore, the first aim of this paper is to show that (8) holds for𝜌>0small when𝑝∈(2,3)by using the standard scaling arguments. To achieve this aim, we introduce a new subset𝐵_𝜌∩𝒫of𝐵_𝜌(see details in Section 3), then we consider the minimization problem (7) constrained on𝐵_𝜌∩𝒫instead of𝐵_𝜌, and we use the standard scaling arguments to prove that (8) holds for𝜌>0small. Moreover, we can get a specific estimate on𝜌that allows us to obtain the sign and the behavior of the Lagrange multiplier𝜆_𝜌if𝑝∈(2,8/3]; these are not considered in [18].

The other aim of this paper is to study the properties of the Lagrange multiplier𝜆_𝜌and the ratio𝐼_𝜌2/𝜌²corresponding to the solution(𝑢_𝜌,𝜆_𝜌)of (1) with‖𝑢_𝜌‖₂=𝜌. It is known that𝜆_𝜌and𝐼_𝜌2/𝜌²are 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 in𝐻¹(ℝ³;ℂ)up to translations provided that one of the following conditions holds (1)𝑝∈(2,8/3]and0<𝜌<𝜌₁∶=3^{(3𝑝−8)/8(3−𝑝)}𝜋^{(3𝑝−10)/8(3−𝑝)}𝑆^3/2((𝑝−2)/2𝑝)^{1/4(3−𝑝)}, where𝑆is defined by (12); (2)𝑝∈(8/3,3)and0<𝜌<𝜌₂for some𝜌₂>0. In particular, (1) has a solution(𝑢_𝜌,𝜆_𝜌)∈𝐻¹(ℝ³;ℂ)×ℝsuch that𝐼(𝑢_𝜌)=𝐼_𝜌2and‖𝑢_𝜌‖₂=𝜌. Moreover, if the above assumption (1) holds and(𝑢_𝜌,𝜆_𝜌)is a solution of (1) with‖𝑢_𝜌‖₂=𝜌>0and𝐼(𝑢_𝜌)=𝐼_𝜌2, then𝜆_𝜌<0,𝜆_𝜌→0and𝐼_𝜌2/𝜌²→0as𝜌→0, respectively.

Theorem 2. Let𝑝∈(2,12/5]and let 𝜌>0. If(𝑢_𝜌,𝜆_𝜌)is a solution of (1) with‖𝑢_𝜌‖₂=𝜌, then we have (i)𝜆_𝜌<0,𝐼(𝑢_𝜌)<0,𝜆_𝜌→0as𝜌→0and there exists a positive constant𝐶₁, independent of𝜌, such that𝜆_𝜌∈(−𝐶₁,0); (ii)there exists a positive constant𝐶₂, independent of𝜌, such that𝐼(𝑢_𝜌)/𝜌²∈(−𝐶₂,0). In particular, if𝐼(𝑢_𝜌)=𝐼_𝜌2, then𝐼_𝜌2/𝜌²∈(−𝐶₂,0).

Remarks. (a) We point out that parts of Theorem 1 are already contained in [18, Theorem 4.1]. In the proof of Theorem 1, with𝜌₁in hand, we can obtain some additional information of the Lagrange multiplier𝜆_𝜌and the ratio𝐼_𝜌2/𝜌²when𝑝∈(2,8/3], and these are not contained in [18, Theorem 4.1]. However, we do not know whether𝜌₁is optimal or not.

(b) Theorem 2(i)shows that (1) has only the zero solution if𝑝∈(2,12/5]and𝜆≥0. In the case of𝑝∈(2,3), it was shown in [5, 20] (see also [13, Remark 1.4]) that there exists𝜆₀<0such that (1) has only the zero solution for𝜆∈(−∞,𝜆₀). The nonexistence results of nonzero solutions of (1) were also discussed in [13] for𝑝∈[3,10/3].

(c) As we have anticipated, the existence of minimizers for𝐼_𝜌2is 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 of𝐵_𝜌 and then analyze the properties of minimizing sequences for𝐼_𝜌2constrained on the new subset, and finally, we prove that (8) holds when𝑝∈(2,8/3]and𝑝∈(8/3,3), respectively.

2. Preliminaries

Throughout this paper, all the functions, unless otherwise stated, are complex valued, but for simplicity we will write𝐿^𝑞(ℝ³),𝐻¹(ℝ³)and𝒟^1,2(ℝ³)defined in the following: (i)𝐿^𝑞(ℝ³)is the usual Lebesgue space endowed with the norm‖𝑢‖_𝑞∶=(∫_ℝ3|𝑢|^𝑞𝑑𝑥)^1/𝑞, where𝑞∈[1,∞); (ii)𝐻¹(ℝ³)is the usual Sobolev space endowed with the norm ‖𝑢‖∶=_ℝ3|∇𝑢|²+|𝑢|²𝑑𝑥^1/2; (10)(iii)𝒟^1,2(ℝ³)is the completion of𝐶^∞₀(ℝ³)with respect to the norm ‖∇𝑢‖₂=_ℝ3|∇𝑢|²𝑑𝑥^1/2; (11)(iv)𝑆is the best Sobolev imbedding constant of𝒟^1,2(ℝ³)↪𝐿⁶(ℝ³)defined as 𝑆∶=inf‖∇𝑢‖₂∶𝑢∈𝒟^1,2ℝ³,‖𝑢‖₆=1. (12)Moreover, the letter𝐶will denote a suitable positive constant, whose value may change in the same line, and the symbol𝑜(1)denotes a quantity which goes to zero. We also use𝑂(1)to denote a bounded quantity.

Let𝜙_𝑢(𝑥)=|𝑥−𝑦|⁻¹∗|𝑢|², and then, for each𝑢∈𝐻¹(ℝ³),𝜙_𝑢is 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], _ℝ3∇𝜙_𝑢²𝑑𝑥=4𝜋_ℝ3𝜙_𝑢|𝑢|²𝑑𝑥=4𝜋_ℝ3|𝑢(𝑥)|²𝑢𝑦²𝑥−𝑦𝑑𝑥𝑑𝑦,_ℝ3𝜙_𝑢|𝑢|²𝑑𝑥≤𝜙_𝑢6‖𝑢‖²_12/5≤4𝜋𝑆²‖𝑢‖⁴_12/5. (13)(14) For each𝜌>0, let𝑢∈𝐵_𝜌and𝑢^𝑡(𝑥)=𝑡^3/2𝑢(𝑡𝑥)(𝑡>0), and we have‖𝑢^𝑡‖₂=‖𝑢‖₂=𝜌, that is,𝑢^𝑡∈𝐵_𝜌. Let 𝑓_𝑢(𝑡)=𝐼𝑢^𝑡=𝑡²2_ℝ³|∇𝑢|²𝑡𝑑𝑥+4_ℝ³𝜙_𝑢|𝑢|²−𝑡𝑑𝑥_{(3/2)(𝑝−2)}𝑝_ℝ³|𝑢|^𝑝𝑑𝑥; (15) it is clear that𝐼_𝜌2≤0for all𝜌>0since𝑓_𝑢(𝑡)→0as𝑡→0.

We now recall an abstract result on the constrained minimization problem 𝐽_𝜌2=inf_{𝑢∈𝐵𝜌}𝐽(𝑢),weagree𝐽₀=0, (16) where𝜌>0,𝐵_𝜌={𝑢∈𝐻¹(ℝ³)∶‖𝑢‖₂=𝜌},𝐽_𝜌2>−∞is assumed, and 𝐽(𝑢)=12‖∇𝑢‖²₂+𝑇(𝑢), (17) for some functional𝑇∈𝐶¹(𝐻¹(ℝ³),ℝ).

Lemma 3 (see [17, 18]). Let𝑇∈𝐶¹(𝐻¹(ℝ³),ℝ). Let𝜌>0and{𝑢_𝑛}⊂𝐵_𝜌be a minimizing sequence for𝐽_𝜌2weakly convergent, up to translations, to a nonzero function𝑢. Assume that (8) holds and that 𝑇𝑢_𝑛−𝑢+𝑇(𝑢)=𝑇𝑢_𝑛+𝑜(1);𝑇𝛼_𝑛𝑢_𝑛−𝑢−𝑇𝑢_𝑛−𝑢=𝑜(1)𝑤ℎ𝑒𝑟𝑒𝛼_𝑛=𝜌²−‖𝑢‖²₂𝑢_𝑛−𝑢₂;⟨𝑇^′𝑢_𝑛,𝑢_𝑛⟩=𝑂(1);𝑇^′𝑢_𝑛−𝑇^′𝑢_𝑚,𝑢_𝑛−𝑢_𝑚=𝑜(1)𝑎𝑠𝑛,𝑚⟶∞. (18)(19)(20)(21) Then‖𝑢_𝑛−𝑢‖→0. In particular it follows that𝑢∈𝐵_𝜌and𝐽(𝑢)=𝐽_𝜌2.

As pointed out in [18], Lemma 3 is a variant of the concentration-compactness principle of Lions [15, 16]. Assumption (18) shows that𝑇possesses the Brizis-Lieb splitting property and (19) is the homogeneity of𝑇. If, in addition, the condition (8) holds, then one can show that dichotomy does not occur; that is,𝑢∈𝐵_𝜌. Furthermore, if (20) and (21) are also fulfilled, then{𝑢_𝑛}strongly converges to𝑢in𝐻¹(ℝ³). Finally we recall the following results obtained in [17, 18].

Lemma 4 (see [18]). If𝑝∈(2,3), then𝐼_𝜌2<0for all𝜌>0.

Lemma 5 (see [17, Lemma 3.1]). If𝑝∈(2,10/3), then, for every𝜌>0, the functional𝐼is bounded below and coercive on𝐵_𝜌.

Remark 6. For𝑝∈(2,3), it follows from Lemmas 4 and 5 that each minimizing sequence for𝐼_𝜌2is bounded from below and above by two positive constants in𝒟^1,2(ℝ³)and𝐻¹(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 𝒫∶=𝑢∈𝐻¹ℝ³∶𝑄(𝑢)=0,𝐼(𝑢)=min_𝑡>0𝐼𝑢^𝑡, (22) where𝑢^𝑡(𝑥)=𝑡^3/2𝑢(𝑡𝑥)with𝑡>0and𝑄(𝑢)is a functional on𝐻¹(ℝ³)defined as 𝑄(𝑢)=_ℝ3|∇𝑢|²1𝑑𝑥+4_ℝ³𝜙_𝑢|𝑢|²−3𝑑𝑥𝑝−22𝑝_ℝ³|𝑢|^𝑝𝑑𝑥. (23) It was shown in [13, Lemma 2.1] that if𝑢_𝜌is a constrained critical point of𝐼on𝐵_𝜌associated with the Lagrange multiplier𝜆_𝜌,then𝑄(𝑢_𝜌)=0, which is nothing but a linear combination of⟨𝐸^′(𝑢_𝜌),𝑢_𝜌⟩=0(recall that𝐸(𝑢)is given by (3)) and the following Pohozaev identity for (1) (cf. [5, 9]) 12∇𝑢_𝜌²₂−3𝜆_𝜌2𝑢_𝜌²₂+54_ℝ³𝜙_𝑢𝜌𝑢_𝜌²3𝑑𝑥−𝑝𝑢_𝜌^𝑝_𝑝=0. (24) The following lemma shows that𝐵_𝜌∩𝒫is well defined.

Lemma 7. Let𝑝∈(2,3)and let 𝜌>0. For each𝑢∈𝐵_𝜌with𝐼(𝑢)<0, there exists a unique𝑡_𝑢>0such that𝐼(𝑢^𝑡𝑢)=min{𝐼(𝑢^𝑡)∶𝑡>0}; moreover,𝑢^𝑡𝑢∈𝐵_𝜌∩𝒫.

Proof. We divide the proof into two cases.
Case   1 (𝑝∈(2,8/3)). Let𝑢∈𝐵_𝜌, for simplicity, and we will write𝑓^′_𝑢(𝑡),𝑓_𝑢^′′(𝑡)and𝑓_𝑢^′′′(𝑡), the derivatives of𝑓_𝑢(𝑡)on𝑡, instead of𝑑𝑓_𝑢(𝑡)/𝑑𝑡,𝑑²𝑓_𝑢(𝑡)/𝑑𝑡²and𝑑³𝑓_𝑢(𝑡)/𝑑𝑡³. From (15), we have 𝑓^′_𝑢(𝑡)=𝑡_ℝ3|∇𝑢|²1𝑑𝑥+4_ℝ³𝜙_𝑢|𝑢|²−3𝑑𝑥𝑝−2𝑡2𝑝^{(3𝑝−8)/2}_ℝ³|𝑢|^𝑝𝑑𝑥. (25) Noting that(3𝑝−8)/2∈(−1,0)since𝑝∈(2,8/3), then, by (25),lim_𝑡→0𝑓^′_𝑢(𝑡)=−∞andlim_𝑡→∞𝑓^′_𝑢(𝑡)=+∞; thus there exists𝑡_𝑢>0such that𝑓^′_𝑢(𝑡_𝑢)=0. If there exists another𝑠_𝑢>0such that𝑓^′_𝑢(𝑠_𝑢)=0, without loss of generality, we may assume that𝑠_𝑢>𝑡_𝑢, and then we get 0=𝑓^′_𝑢s_𝑢−𝑓^′_𝑢𝑡_𝑢=𝑠_𝑢−𝑡_𝑢ℝ3|∇𝑢|²+3𝑑𝑥𝑝−2𝑡2𝑝^{𝑢(3𝑝−8)/2}−𝑠_𝑢^{(3𝑝−8)/2}_ℝ3|𝑢|^𝑝𝑑𝑥>0, (26) a contradiction. Therefore,𝑡_𝑢is unique and it is clear that𝐼(𝑢^𝑡𝑢)=min{𝐼(𝑢^𝑡)∶𝑡>0}. Moreover,𝑢^𝑡𝑢∈𝐵_𝜌∩𝒫because of𝑓^′_𝑢(𝑡_𝑢)𝑡_𝑢=0.
Case  2  (𝑝∈[8/3,3)). By Lemma 4, we know that the set𝐴_𝜌∶={𝑢∈𝐵_𝜌∶𝐼(𝑢)<0}≠∅. Let𝑢∈𝐴_𝜌, if𝑓^′_𝑢(𝑡)>0for all𝑡>0; that is,𝑓_𝑢(𝑡)is strictly increasing, then we obtain that𝑓_𝑢(𝑡)<𝑓_𝑢(1)=𝐼(𝑢)<0for all𝑡∈(0,1). However, it is easy to see thatlim_𝑡→0𝑓_𝑢(𝑡)=0; this is a contradiction. On the other hand, we know that𝑓_𝑢(𝑡)→∞as𝑡→∞; hence there is a𝑡_𝑢>0such that𝑓^′_𝑢(𝑡_𝑢)=0,𝑢^𝑡𝑢∈𝐵_𝜌∩𝒫and 𝑓_𝑢𝑡_𝑢=min𝑓_𝑢(𝑡)∶𝑡>0≤𝑓_𝑢(1)=𝐼(𝑢)<0. (27) Next, we will show that𝑡_𝑢is unique. Arguing by contradiction, suppose that there is another𝑠_𝑢>0such that𝑓_𝑢(𝑡_𝑢)=𝑓_𝑢(𝑠_𝑢)=min{𝑓_𝑢(𝑡)∶𝑡>0}, without loss of generality, we may assume that𝑠_𝑢>𝑡_𝑢, and then we have 𝑓^′_𝑢𝑡_𝑢=𝑓^′_𝑢𝑠_𝑢=0,𝑓_𝑢^′′𝑡_𝑢≥0,𝑓_𝑢^′′𝑠_𝑢≥0. (28) According to (28), there exists𝜔_𝑢∈(𝑡_𝑢,𝑠_𝑢)such that𝑓_𝑢^′′(𝜔_𝑢)=0. After a simple calculation, we get 𝑓_𝑢^′′(𝑡)=‖∇𝑢‖²₂−3𝑝−23𝑝−8𝑡4𝑝^{(3𝑝−10)/2}_ℝ³|𝑢|^𝑝𝑑𝑥,𝑓_𝑢^′′′(𝑡)3=−𝑝−23𝑝−83𝑝−10𝑡8𝑝^{(3𝑝−12)/2}×_ℝ3|𝑢|^𝑝𝑑𝑥. (29)(30) If𝑝=8/3,then, by (29),𝑓_𝑢^′′(𝑡)>0for all𝑡>0, which contradicts𝑓_𝑢^′′(𝜔_𝑢)=0. If𝑝∈(8/3,3), then, by (30),𝑓_𝑢^′′′(𝑡)>0for all𝑡>0. Noting that𝜔_𝑢∈(𝑡_𝑢,𝑠_𝑢), we have 0≤𝑓_𝑢^′′𝑡_𝑢<𝑓_𝑢^′′𝜔_𝑢=0, (31) again a contradiction. Therefore,𝑡_𝑢>0is unique.

Lemma 8. Let𝑝∈(2,3)and𝜌>0. For each{𝑢_𝑛}⊂𝐵_𝜌such that𝐼(𝑢_𝑛)→𝐼_𝜌2<0as𝑛→∞and𝐼(𝑢_𝑛)<0for all𝑛∈ℕ, there exists a bounded sequence{𝑡_𝑛}⊂ℝ⁺such that{𝑢^𝑡𝑛_𝑛}⊂𝐵_𝜌∩𝒫and𝐼(𝑢^𝑡𝑛_𝑛)→𝐼_𝜌2as𝑛→∞with𝐼(𝑢^𝑡𝑛_𝑛)<0for all𝑛∈ℕ; that is,{𝑢^𝑡𝑛_𝑛}is also a minimizing sequence for𝐼_𝜌2constrained on𝐵_𝜌.

Proof. It follows from Lemma 7 that, for each𝑢_𝑛, there exists𝑡_𝑛>0such that𝑢^𝑡𝑛_𝑛∈𝐵_𝜌∩𝒫and𝐼(𝑢^𝑡𝑛_𝑛)=min{𝐼(𝑢^𝑡_𝑛)∶𝑡>0}≤𝐼(𝑢_𝑛)<0; therefore, we have 𝐼_𝜌2≤𝐼𝑢^𝑡𝑛_𝑛≤𝐼𝑢_𝑛⟶𝐼_𝜌2, (32) as𝑛→∞, that is,{𝑢^𝑡𝑛_𝑛}is a minimizing sequence. Next, we will show that{𝑡_𝑛}is bounded. Indeed, from Remark 6,{𝑢_𝑛}and{𝑢^𝑡𝑛_𝑛}are bounded from below and above by two positive constants in𝒟^1,2(ℝ³)and𝐻¹(ℝ³), respectively. Noting that∫_ℝ3|∇𝑢^𝑡𝑛_𝑛|²𝑑𝑥=𝑡²_𝑛∫_ℝ3|∇𝑢_𝑛|²𝑑𝑥; therefore,{𝑡_𝑛}is bounded from below and above by two positive constants.

Remark 9. Thanks to the Lemma 8, we know that𝐼_𝜌2=inf{𝐼(𝑢)∶𝑢∈𝐵_𝜌}=inf{𝐼(𝑢)∶𝑢∈𝐵_𝜌∩𝒫}, and, in the following, we will consider the minimization problem (7) restricted to𝐵_𝜌∩𝒫instead of𝐵_𝜌. By Lemmas 4 and 8, for each𝜌>0, if{𝑢_𝑛}⊂𝐵_𝜌∩𝒫satisfying𝐼(𝑢_𝑛)→𝐼_𝜌2as𝑛→∞, then, up to a subsequence, we may assume that𝐼(𝑢_𝑛)<0. It follows from Lemma 5 that{𝑢_𝑛}is bounded in𝐻¹(ℝ³); by the results of [17, 18], we may assume that𝑢_𝑛⇀𝑢≠0as𝑛→∞in𝐻¹(ℝ³).
The following estimates of the elements of𝐵_𝜌∩𝒫are crucial to proving the strong subadditivity inequality (8).

Lemma 10. Let𝑝∈(2,3)and𝜌>0. For each𝑢∈𝐵_𝜌∩𝒫, it holds ‖∇𝑢‖²₂≤3𝑝−22𝑝^{4/(10−3𝑝)}𝑆^{−6(𝑝−2)/(10−3𝑝)}𝜌^{2(6−𝑝)/(10−3𝑝)},‖𝑢‖^𝑝_𝑝≤3𝑝−22𝑝^{3(𝑝−2)/(10−3𝑝)}𝑆^{−6(𝑝−2)/(10−3𝑝)}𝜌^{2(6−𝑝)/(10−3𝑝)}. (33)

Proof. Since𝑢∈𝐵_𝜌∩𝒫, ‖∇𝑢‖²₂+14_ℝ³𝜙_𝑢|𝑢|²3𝑑𝑥−𝑝−2‖𝑢‖2𝑝^𝑝𝑝=0. (34) Noting that∫_ℝ3𝜙_𝑢|𝑢|²𝑑𝑥≥0(see (13)), by using the Hölder inequality, we get ‖∇𝑢‖²₂≤3𝑝−2‖𝑢‖2𝑝^𝑝𝑝≤3𝑝−2‖𝑢‖2𝑝^{2(6−𝑝)/2}‖𝑢‖₆^{3(𝑝−2)/2}≤3𝑝−2𝑆2𝑝^{−3(𝑝−2)/2}‖𝑢‖₂^(6−𝑝)/2‖∇𝑢‖₂^{3(𝑝−2)/2}, (35) which implies that ‖∇𝑢‖²₂≤3𝑝−22𝑝^{4/(10−3𝑝)}𝑆^{−6(𝑝−2)/(10−3𝑝)}𝜌^{2(6−𝑝)/(10−3𝑝)}. (36) On the other hand, we have ‖𝑢‖^𝑝_𝑝≤‖𝑢‖₂^(6−𝑝)/2‖𝑢‖₆^{3(𝑝−2)/2}≤𝑆^{−3(𝑝−2)/2}‖𝑢‖₂^(6−𝑝)/2‖∇𝑢‖₂^{3(𝑝−2)/2}≤3𝑝−22𝑝^{3(𝑝−2)/(10−3𝑝)}×𝑆^{−6(𝑝−2)/(10−3𝑝)}𝜌^{2(6−𝑝)/(10−3𝑝)}; (37) this concludes the proof of this lemma.

Remark 11. Let𝑝∈(3,10/3). It was shown in [13, Theorem 1.1] that𝐼_𝜌2<0if and only if𝜌∈(𝜌,∞), where the positive number𝜌is defined as 𝜌=inf𝜌>0∶𝐼_𝜌2<0. (38) Therefore, after a simple calculation, we can show that both of Lemmas 7 and 10 hold if𝑝∈(3,10/3)and𝜌∈(𝜌,∞).
Motivated by [17], we will use the standard scaling arguments to prove that the strong subadditivity inequality (8) holds for𝑝∈(2,3). First, we consider the case of𝑝∈(2,8/3].

Lemma 12. For𝑝∈(2,8/3], let 𝜌₁=3^{(3𝑝−8)/8(3−𝑝)}𝜋^{(3𝑝−10)/8(3−𝑝)}𝑆^3/2𝑝−22𝑝^{1/4(3−𝑝)}>0. (39) Then 𝐼_𝜌2<𝐼_𝜇2+𝐼_{𝜌2−𝜇2}∀0<𝜇<𝜌<𝜌₁. (40)

Proof. By Lemma 8 and Remark 9, for each{𝑢_𝑛}⊂𝐵_𝜌∩𝒫satisfying𝐼(𝑢_𝑛)→𝐼_𝜌2<0as𝑛→∞, we may assume that, for all𝑛,𝐼(𝑢_𝑛)≤𝐼_𝜌2/2, which implies that 𝑢_𝑛^𝑝_𝑝𝑝≥−2𝐼_𝜌2. (41) Noting that𝑡𝑢_𝑛∈𝐵_𝑡𝜌(𝑡>0), we have 𝐼𝑡𝑢_𝑛=𝑡²2_ℝ³∇𝑢_𝑛²𝑡𝑑𝑥+⁴4_ℝ³𝜙_𝑢𝑛𝑢_𝑛²−𝑡𝑑𝑥_𝑝𝑝_ℝ³𝑢_𝑛^𝑝𝑑𝑥=𝑡²𝐼𝑢_𝑛+𝑡²−14_ℝ³𝜙_𝑢𝑛𝑢_𝑛²−𝑡𝑑𝑥_𝑝−2−1𝑝_ℝ³𝑢_𝑛^𝑝𝑑𝑥=𝑡²𝐼𝑢_𝑛+𝑔𝑡,𝑢_𝑛, (42) where 𝑔()=𝑡𝑡,𝑢²−14_ℝ³𝜙_𝑢|𝑢|²𝑡𝑑𝑥−^𝑝−2−1𝑝_ℝ³|𝑢|^𝑝𝑑𝑥. (43) We calculate the derivative of𝑔(𝑡,𝑢)on𝑡: 𝑑𝑔(𝑡,𝑢)=𝑡𝑑𝑡2_ℝ³𝜙_𝑢|𝑢|²𝑑𝑥−𝑝−2𝑝𝑡^𝑝−3‖𝑢‖^𝑝_𝑝. (44) Letting𝑑𝑔(𝑡,𝑢)/𝑑𝑡=0, we see from (14) that 𝑡^𝑝−4=𝑝∫_ℝ3𝜙_𝑢|𝑢|²𝑑𝑥2𝑝−2‖𝑢‖^𝑝_𝑝≤2𝜋𝑝‖𝑢‖⁴_12/5𝑆²𝑝−2‖𝑢‖^𝑝_𝑝. (45) Furthermore, 𝑑²𝑔(𝑡,𝑢)𝑑𝑡²=12_ℝ³𝜙_𝑢|𝑢|²𝑑𝑥−𝑝−2𝑝−3𝑝𝑡^𝑝−4‖𝑢‖^𝑝_𝑝>0. (46)
Now we divide the value of 𝑝 into two cases to discuss𝑑𝑔(𝑡,𝑢_𝑛)/𝑑𝑡.
Case  1 (𝑝∈(2,12/5)). It follows from Lemma 10, (14), and the Hölder inequality that 𝑑𝑔𝑡,𝑢_𝑛𝑑𝑡_𝑡=1=12_ℝ³𝜙_𝑢𝑛𝑢_𝑛²𝑑𝑥−𝑝−2𝑝_ℝ³𝑢_𝑛^𝑝≤𝑑𝑥2𝜋𝑆²𝑢_𝑛⁴_12/5−𝑝−2𝑝𝑢_𝑛^𝑝_𝑝≤2𝜋𝑆²𝑢_𝑛𝑝^{6𝑝/(6−𝑝)}𝑢_𝑛6^{2(12−5𝑝)/(6−𝑝)}−𝑝−2𝑝𝑢_𝑛^𝑝_𝑝=𝑢_𝑛^𝑝_𝑝2𝜋𝑆^{12(𝑝−3)/(6−𝑝)}∇𝑢_𝑛2^{2(12−5𝑝)/(6−𝑝)}×𝑢_𝑛^𝑝2_𝑝^/(6−𝑝)−𝑝−2𝑝≤𝑢_𝑛^𝑝_𝑝2𝜋𝑆^{12(𝑝−3)/(10−3𝑝)}3𝑝−22𝑝^{(8−3𝑝)/(10−3𝑝)}×𝜌^{8(3−𝑝)/(10−3𝑝)}−𝑝−2𝑝. (47)
Case  2 (𝑝∈[12/5,8/3]). Again by Lemma 10, (14), and the Hölder inequality, we have 𝑑𝑔𝑡,𝑢_𝑛𝑑𝑡_𝑡=1=12_ℝ³𝜙_𝑢𝑛𝑢_𝑛²𝑑𝑥−𝑝−2𝑝_ℝ³𝑢_𝑛^𝑝≤𝑑𝑥2𝜋𝑆²𝑢_𝑛⁴_12/5−𝑝−2𝑝𝑢_𝑛^𝑝_𝑝≤2𝜋𝑆²𝑢_𝑛2^{2(5𝑝−12)/3(𝑝−2)}𝑢_𝑛𝑝^{2𝑝/3(𝑝−2)}−𝑝−2𝑝𝑢_𝑛^𝑝_𝑝≤𝑢_𝑛^𝑝_𝑝×2𝜋𝑆²𝑢_𝑛2^{2(5𝑝−12)/3(𝑝−2)}𝑢_𝑛𝑝^{𝑝(8−3𝑝)/3(𝑝−2)}−𝑝−2𝑝≤𝑢_𝑛^𝑝_𝑝2𝜋𝑆^{12(𝑝−3)/(10−3𝑝)}3𝑝−22𝑝^{(8−3𝑝)/(10−3𝑝)}×𝜌^{8(3−𝑝)/(10−3𝑝)}−𝑝−2𝑝. (48) Let 𝜌₁=3^{(3𝑝−8)/8(3−𝑝)}𝜋^{(3𝑝−10)/8(3−𝑝)}𝑆^3/2𝑝−22𝑝^{1/4(3−𝑝)}. (49) Then by (47), (48), and (49), we know that, for each0<𝜌<𝜌₁, there holds 𝜌₁𝜌∶=2𝜋𝑝𝑆𝑝−2^{12(𝑝−3)/(10−3𝑝)}×3𝑝−22𝑝^{(8−3𝑝)/(10−3𝑝)}𝜌^{8(3−𝑝)/(10−3𝑝)}<1,𝑑𝑔𝑡,𝑢_𝑛𝑑𝑡_𝑡=1<0, (50) for all𝑛∈ℕ. On the other hand, for each𝜀∈(0,1−𝜌₁(𝜌)), it follows from (41), (44), (45), (46), and Lemma 10 that, for all𝑡∈(1,(𝜌₁(𝜌)/(1−𝜀))^1/(𝑝−4))and all𝑛∈ℕ, 𝑑𝑔𝑡,𝑢_𝑛=𝑑𝑡𝑝−2𝑝𝑢_𝑛^𝑝_𝑝𝑡^𝑝−3𝑝∫_ℝ3𝜙_𝑢𝑛𝑢_𝑛²𝑑𝑥2𝑝−2𝑢_𝑛^𝑝_𝑝𝑡^4−𝑝≤𝜀−1𝑝−22𝐼_𝜌2<0. (51) This, together with the mean value theorem and (41), yields that for all𝑡∈(1,((1−𝜀)⁻¹𝜌₁(𝜌))^1/(𝑝−4))and all𝑛∈ℕ, 𝑔𝑡,𝑢_𝑛=𝑔1,𝑢_𝑛+𝑑𝑔𝑡,𝑢_𝑛𝑑𝑡_{𝑡=𝜃𝑡𝑛}(𝑡−1)<−𝐶(𝑡−1), (52) where𝜃_𝑡𝑛∈(1,𝑡)and𝐶>0depend only on𝜀,𝑝,and𝜌. By (42), we have 𝐼_(𝑡𝜌)²≤𝐼𝑡𝑢_𝑛=𝑡²𝐼𝑢_𝑛+𝑔𝑡,𝑢_𝑛≤𝑡²𝐼𝑢_𝑛−𝐶𝑡²(𝑡−1)=𝑡²𝐼_𝜌2−𝐶𝑡²(𝑡−1)+𝑜(1), (53) then 𝐼_(𝑡𝜌)²<𝑡²𝐼_𝜌2𝜌∀𝑡∈1,¹𝜌1−𝜀^1/(𝑝−4),𝜌∈0,𝜌₁. (54) Clearly,𝜌₁(cf. (50)) is strictly increasing on𝜌, and then(𝜌₁/(1−𝜀))^1/(𝑝−4)is strictly decreasing on𝜌since𝑝∈(2,3).
Let ℎ𝜌𝜌∶=¹𝜌1−𝜀^1/(𝑝−4). (55) For each𝜌∈(0,𝜌₁), let𝜇∈(0,𝜌) without loss of generality, we may assume that𝜇>𝜌²−𝜇². Choosing𝜀∈(0,min{𝜌₁−𝜌,1−𝜌₁(𝜌),𝜇−𝜌²−𝜇²}), then by (50) we know thatℎ(𝜌²−𝜇²)>ℎ(𝜇)>ℎ(𝜌)>1. (a)If𝜌/𝜇∈(1,ℎ(𝜇)), then by (54) 𝐼_𝜌2=𝐼_{(𝜌2/𝜇2)𝜇2}<𝜌²𝜇²𝐼_𝜇2=𝐼_𝜇2+𝜌²−𝜇²𝜇²𝐼_𝜇2=𝐼_𝜇2+𝜌²−𝜇²𝜇²𝐼_(𝜇²_/(𝜌²_{−𝜇2))(𝜌2−𝜇2)}. (56)(b)If𝜌/𝜇∉(1,ℎ(𝜇)), then there exists𝑘∈ℕsuch that(𝜌/𝜇)^1/𝑘∈(1,ℎ(𝜌)). Therefore 𝜌𝜇^1/𝑘∈𝜌1,ℎ𝜇^{(𝑘−𝑖)/𝑘}𝜇,∀𝑖=1,2,…,𝑘. (57)
It follows from (54) that 𝐼_𝜌2=𝐼_(𝜌/𝜇)^2/𝑘_((𝜌/𝜇)^{2(𝑘−1)/𝑘}_𝜇2)<𝜌𝜇^2/𝑘𝐼_(𝜌/𝜇)^{2(𝑘−1)/𝑘}_𝜇2𝜌<⋯<²𝜇²𝐼_𝜇2. (58) Combining the above cases (a) and (b), we can show that 𝐼_𝜌2<𝐼_𝜇2+𝐼_{𝜌2−𝜇2}∀0<𝜇<𝜌<𝜌₁. (59) Thus the conclusion of this lemma holds.

Remark 13. For the case of𝑝=8/3, it has been proved in [4, 17] that the strong subadditivity inequality (8) holds for𝜌>0small. By using the result of [17], we can give a specific estimate of lower bound of𝜌such that (8) holds; that is, (8) holds for all𝜌∈(0,(8𝜋)^−3/4𝑆^3/2). However, if we plug𝑝=8/3into (49), then we have𝜌₁=(8𝜋)^−3/4𝑆^3/2, which coincides with the one given in [17].

Next, we will show (8) for𝑝∈(8/3,3). We point out that the case of𝑝∈(8/3,3)is quite different from the case of𝑝∈(2,8/3]since the inequality (48) does not hold anymore. Inspired by [18], we will give some estimates for𝐼_𝜌2in Lemmas 14 and 15, and these are crucial for the proof of (8) if𝑝∈(8/3,3).

Lemma 14. Let𝑝∈(8/3,3)and𝜌>0be fixed. If there exists𝑢∈𝐵_𝜌∩𝒫such that𝐼(𝑢)≤𝐼_𝜌2/2and ‖𝑢‖^𝑝_𝑝>3‖∇𝑢‖²₂, (60) then there exist positive constants𝐶₃and𝐶₄dependent on𝑝and𝜌, such that 𝐼_𝜇2≤−𝐶₃𝜇^{2(6−𝑝)/(10−3𝑝)}+𝐶₄𝜇^{2(18−5𝑝)/(10−3𝑝)}∀𝜇>0. (61)

Proof. From the assumptions of the lemma, we see that ‖∇𝑢‖²₂+14_ℝ³𝜙_𝑢|𝑢|²3𝑑𝑥−𝑝−2‖𝑢‖2𝑝^𝑝𝑝=0,12‖∇𝑢‖²₂+14_ℝ³𝜙_𝑢|𝑢|²1𝑑𝑥−𝑝‖𝑢‖^𝑝_𝑝(𝑢)≤𝐼=𝐼^𝜌22. (62)(63) By (60), (62), and (63), we deduce that 𝐼_𝜌22(𝑢)1≥𝐼=−2‖∇𝑢‖²₂+3𝑝−8‖𝑢‖2𝑝^𝑝𝑝>4𝑝−3𝑝‖∇𝑢‖²₂. (64) Combining (62) and (64), and using Lemma 10, we also obtain 14_ℝ³𝜙_𝑢|𝑢|²=3𝑑𝑥𝑝−2‖𝑢‖2𝑝^𝑝𝑝−‖∇𝑢‖²₂≤3𝑝−22𝑝^{4/(10−3𝑝)}𝑆^{−6(𝑝−2)/(10−3𝑝)}×𝜌^{2(6−𝑝)/(10−3𝑝)}+𝑝8𝐼3−𝑝^𝜌2. (65) For each𝑡>0, let𝑢_𝑡(𝑥)=𝑡^{4/(10−3𝑝)}𝑢(𝑡^{2(𝑝−2)/(10−3𝑝)}𝑥), we have‖𝑢_𝑡‖₂=𝑡‖𝑢‖₂=𝑡𝜌. It follows from (60), (64), and (65) that 𝐼_(𝑡𝜌)²≤𝐼𝑢_𝑡=12𝑡^{2(6−𝑝)/(10−3𝑝)}‖∇𝑢‖²₂+14𝑡^{2(18−5𝑝)/(10−3𝑝)}_ℝ3𝜙_𝑢|𝑢|²−1𝑑𝑥𝑝𝑡^{2(6−𝑝)/(10−3𝑝)}‖𝑢‖^𝑝_𝑝≤12−3𝑝𝑡^{2(6−𝑝)/(10−3𝑝)}‖∇𝑢‖²₂+14𝑡^{2(18−5𝑝)/(10−3𝑝)}_ℝ3𝜙_𝑢|𝑢|²≤𝑑𝑥6−𝑝𝐼163−𝑝𝜌^{𝜌22(6−𝑝)/(10−3𝑝)}𝑡𝜌^{2(6−𝑝)/(10−3𝑝)}+3𝑝−22𝑝^{4/(10−3𝑝)}𝑆^{−6(𝑝−2)/(10−3𝑝)}×𝜌^{8(𝑝−3)/(10−3𝑝)}+𝑝8𝐼3−𝑝𝜌^{𝜌22(18−5𝑝)/(10−3𝑝)}×𝑡𝜌^{2(18−5𝑝)/(10−3𝑝)}. (66) Set𝑡𝜌=𝜇, then𝜇∈(0,∞)since𝑡∈(0,∞)and𝜌is a fixed positive constant. From the above inequality, we see that 𝐼_𝜇2≤−𝐶₃𝜇^{2(6−𝑝)/(10−3𝑝)}+𝐶₄𝜇^{2(18−5𝑝)/(10−3𝑝)}, (67) for some positive constants𝐶₃and𝐶₄depending on𝑝and𝜌.

Lemma 15. Suppose that𝑝∈(8/3,3)and{𝑢_𝑘}⊂𝐵_𝜌𝑘∩𝒫satisfying‖𝑢_𝑘‖^𝑝_𝑝>3‖∇𝑢_𝑘‖²₂and𝐼(𝑢_𝑘)≤𝐼_𝜌²_𝑘/2for all𝑘∈ℕ. Then there exists a positive constant𝐶dependent on𝑝, such that 𝐼_𝜌²_𝑘≥−𝐶𝜌_𝑘^{2(5𝑝−12)/(3𝑝−8)}∀𝑘∈ℕ. (68)

Proof. Following the line of the proof of Lemma 14, we arrive that 14_ℝ³𝜙_𝑢𝑘𝑢_𝑘²3𝑑𝑥=𝑝−2𝑢2𝑝^𝑘𝑝𝑝−∇𝑢_𝑘²₂>7𝑝−18𝑢6𝑝^𝑘𝑝𝑝, (69) which, together with (14) and the Höder inequality, implies that 𝑢_𝑘^𝑝_𝑝≤3𝑝27𝑝−18_ℝ³𝜙_𝑢𝑘𝑢_𝑘²≤𝑑𝑥6𝜋𝑝7𝑝−18𝑆²𝑢_𝑘⁴_12/5≤6𝜋𝑝7𝑝−18𝑆²𝑢_𝑘2^{2(5𝑝−12)/3(𝑝−2)}𝑢_𝑘𝑝^{2𝑝/3(𝑝−2)}, (70) and then 𝑢_𝑘^𝑝_𝑝≤6𝜋𝑝7𝑝−18𝑆^{23(𝑝−2)/(3𝑝−8)}𝑢_𝑘2^{2(5𝑝−12)/(3𝑝−8)}=6𝜋𝑝7𝑝−18𝑆^{23(𝑝−2)/(3𝑝−8)}𝜌_𝑘^{2(5𝑝−12)/(3𝑝−8)}. (71) Combining (62), (69), and (71), we have 𝐼_𝜌²_𝑘2>𝐼𝑢_𝑘=12∇𝑢_𝑘²₂+14_ℝ³𝜙_𝑢𝑘𝑢_𝑘²1𝑑𝑥−𝑝𝑢_𝑘^𝑝_𝑝=3𝑝−10𝑢4𝑝^𝑘𝑝𝑝+18_ℝ³𝜙_𝑢𝑘𝑢_𝑘²≥4𝑑𝑥𝑝−3𝑢3𝑝^𝑘𝑝𝑝≥4𝑝−33𝑝6𝜋𝑝7𝑝−18𝑆^{23(𝑝−2)/(3𝑝−8)}×𝜌_𝑘^{2(5𝑝−12)/(3𝑝−8)}, (72) and this completes the proof.

Lemma 16. If𝑝∈(8/3,3), then there exists a positive constant𝜌₂such that 𝐼_𝜌2<𝐼_𝜇2+𝐼_{𝜌2−𝜇2}∀0<𝜇<𝜌<𝜌₂. (73)

Proof. Suppose that𝜌>0and{𝑢_𝑛}⊂𝐵_𝜌∩𝒫satisfying𝐼(𝑢_𝑛)→𝐼_𝜌2as𝑛→∞. It follows from Remark 9 that, up to a subsequence,𝐼(𝑢_𝑛)≤𝐼_𝜌2/2<0for all𝑛∈ℕ. By Lemma 5, it is easy to see that{𝑢_𝑛}is bounded in𝐻¹(ℝ³). Noting that𝑡𝑢_𝑛∈𝐵_𝑡𝜌, then, by (42), we have 𝐼𝑡𝑢_𝑛=𝑡²𝐼𝑢_𝑛+𝑔𝑡,𝑢_𝑛, (74) where𝑔(𝑡,𝑢)is given by (43). Obviously, 𝑑𝑔𝑡,𝑢_𝑛𝑑𝑡_𝑡=1=12_ℝ³𝜙_𝑢𝑛𝑢_𝑛²𝑑𝑥−𝑝−2𝑝𝑢_𝑛^𝑝_𝑝=𝐼𝑢_𝑛−12∇𝑢_𝑛²₂+14_ℝ³𝜙_𝑢𝑛𝑢_𝑛²𝑑𝑥+3−𝑝𝑝𝑢_𝑛^𝑝_𝑝=𝐼𝑢_𝑛−32∇𝑢_𝑛²₂+12𝑢_𝑛^𝑝_𝑝, (75) since𝑢_𝑛∈𝐵_𝜌∩𝒫and (34) holds. Moreover, 𝑑²𝑔𝑡,𝑢_𝑛𝑑𝑡²=12_ℝ³𝜙_𝑢𝑛𝑢_𝑛²−𝑑𝑥𝑝−2𝑝−3𝑝𝑡^𝑝−4𝑢_𝑛^𝑝_𝑝>0, (76) for all𝑡>0and𝑛∈ℕ.
We claim that there exists𝜌₂>0such that for each𝜌∈(0,𝜌₂)and for each{𝑢_𝑛}⊂𝐵_𝜌∩𝒫satisfying𝐼(𝑢_𝑛)≤𝐼_𝜌2/2<0and𝐼(𝑢_𝑛)→𝐼_𝜌2as𝑛→∞, we have 𝑢_𝑛^𝑝_𝑝≤3∇𝑢_𝑛²₂. (77)
Indeed, if not, we can find{𝜌_𝑘}and{𝑢^𝑘_𝑛}⊂𝐵_𝜌𝑘∩𝒫such that𝜌_𝑘→0as𝑘→∞and for each𝑘∈ℕ,𝐼(𝑢^𝑘_𝑛)→𝐼_𝜌²_𝑘<0as𝑛→∞, but‖𝑢^𝑘_𝑛‖^𝑝_𝑝>3‖∇𝑢^𝑘_𝑛‖²₂. For𝑘=1, there exists𝑛₁>0such that𝐼(𝑢¹_𝑛1)<𝐼_𝜌²₁/2<0, and it can be deduced from Lemma 14 that 𝐼_𝜇2≤−𝐶₃𝜇^{2(6−𝑝)/(10−3𝑝)}+𝐶₄𝜇^{2(18−5𝑝)/(10−3𝑝)}∀𝜇>0, (78) where𝐶₃and𝐶₄are positive constants dependent on𝑝and𝜌₁. On the other hand, we know that for each𝑘∈ℕthere exists𝑛_𝑘>0such that𝐼(𝑢^𝑘_𝑛𝑘)<𝐼_𝜌²_𝑘/2<0. Then by Lemma 15, we obtain 𝐼_𝜌²_𝑘≥−𝐶𝜌_𝑘^{2(5𝑝−12)/(3𝑝−8)}∀𝑘∈ℕ, (79) where𝐶is a positive constant depending only on𝑝. Noting that (78) holds for all𝜇>0, by (78) and (79), we deduce that −𝐶𝜌_𝑘^{2(5𝑝−12)/(3𝑝−8)}≤𝐼_𝜌²_𝑘≤−𝐶₃𝜌_𝑘^{2(6−𝑝)/(10−3𝑝)}+𝐶₄𝜌_𝑘^{2(18−5𝑝)/(10−3𝑝)}, (80) which is a contradiction for𝑘large since𝑝∈(8/3,3)implies 25𝑝−12>23𝑝−86−𝑝.10−3𝑝 (81) Thus we have shown the claim. Now for each𝜌∈(0,𝜌₂)and for all{𝑢_𝑛}⊂𝐵_𝜌∩𝒫with𝐼(𝑢_𝑛)≤𝐼_𝜌2/2and𝐼(𝑢_𝑛)→𝐼_𝜌2as𝑛→∞, using (77), we have 𝑑𝑔𝑡,𝑢_𝑛𝑑𝑡_𝑡=1≤𝐼𝑢_𝑛≤𝐼_𝜌22<0. (82) By (76), similarly as in the proofs of (45) and (51), we get that 𝑡^4−𝑝=2𝑝−2𝑝𝑢_𝑛^𝑝_𝑝∫_ℝ3𝜙_𝑢𝑛𝑢_𝑛²=2𝑑𝑥𝑝−2𝑝‖𝑢_𝑛‖^𝑝_𝑝6𝑝−2/𝑝𝑢_𝑛^𝑝_𝑝−4∇𝑢_𝑛²₂≥3𝑝−27𝑝−18>1. (83) Now, we can choose𝜀>0so small that there exists a positive constant𝐶dependent on𝑝,𝜌,and𝜀, such that 𝑑𝑔𝑡,𝑢_𝑛3()𝑑𝑡≤−𝐶<0∀𝑡∈1,1−𝜀𝑝−27𝑝−18^1/(4−𝑝). (84) Since, for each𝑛,𝑔(1,𝑢_𝑛)=0, it follows that, for each𝑡∈(1,(3(1−𝜀)(𝑝−2)/(7𝑝−18))^1/(4−𝑝)), 𝐼_(𝑡𝜌)²≤𝐼𝑡𝑢_𝑛=𝑡²𝐼𝑢_𝑛+𝑔𝑡,𝑢_𝑛≤𝑡²𝐼𝑢_𝑛+𝑑𝑔𝑡,𝑢_𝑛𝑑𝑡_{𝑡=𝜃𝑡𝑛}(𝑡−1)≤𝑡²𝐼𝑢_𝑛−𝐶(𝑡−1)=𝑡²𝐼_𝜌2−𝐶𝑡²(𝑡−1)+𝑜(1), (85) where𝜃_𝑡𝑛∈(1,𝑡), namely,𝐼_(𝑡𝜌)²<𝑡²𝐼_𝜌2for all𝑡∈(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. Let𝜌>0. Assume that(𝑢_𝜌,𝜆_𝜌)is a solution of (1) with‖𝑢_𝜌‖₂=𝜌. (a)If𝑝∈(2,12/5], then𝜆_𝜌<0. (b)If𝑝∈(12/5,8/3]and𝜆_𝜌≥0, then 𝜌≥𝜌₃∶=6−𝑝6𝜋𝑝^{(10−3𝑝)/8(3−𝑝)}×𝑆^3/23𝑝−22𝑝^{(8−3𝑝)/8(3−𝑝)}. (86)

Proof. Since(𝑢_𝜌,𝜆_𝜌)is a solution of (1), it follows that ∇𝑢_𝜌²₂+_ℝ3𝜙_𝑢𝜌𝑢_𝜌²𝑑𝑥−𝑢_𝜌^𝑝_𝑝=𝜆_𝜌𝑢_𝜌²₂,∇𝑢_𝜌²₂+14_ℝ³𝜙_𝑢𝜌𝑢_𝜌²3𝑑𝑥−𝑝−2𝑢2𝑝^𝜌𝑝𝑝=0. (87)(88) Thus, from (87) and (88), after a simple calculation, we have 𝜆_𝜌𝑢_𝜌²₂=𝑝−63𝑝−2∇𝑢_𝜌²₂+5𝑝−126𝑝−2_ℝ³𝜙_𝑢𝜌𝑢_𝜌²𝑑𝑥, (89) which yields that(a)holds. Moreover, if𝑝∈(12/5,8/3]and𝜆_𝜌≥0, then (89) implies that 5𝑝−126𝑝−2_ℝ³𝜙_𝑢𝜌𝑢_𝜌²𝑑𝑥≥6−𝑝3𝑝−2∇𝑢_𝜌²₂. (90) Thus we get from (88) that 3𝑝−2𝑢2𝑝^𝜌𝑝𝑝=∇𝑢_𝜌²₂+14_ℝ³𝜙_𝑢𝜌𝑢_𝜌²≤9𝑑𝑥𝑝−246−𝑝_ℝ³𝜙_𝑢𝜌𝑢_𝜌²≤9𝜋𝑑𝑥𝑝−26−𝑝𝑆²𝑢_𝜌⁴_12/5. (91) By using the Hölder inequality, it can be deduced from (91) and Lemma 10 that 6−𝑝𝑆6𝜋𝑝²𝑢_𝜌^𝑝_𝑝≤𝑢_𝜌⁴_12/5≤𝑢_𝜌2^{2(5𝑝−12)/3(𝑝−2)}𝑢_𝜌𝑝^{2𝑝/3(𝑝−2)}≤𝑆^{−2(8−3𝑝)/(10−3𝑝)}3𝑝−22𝑝^{(8−3𝑝)/(10−3𝑝)}×𝜌^{8(3−𝑝)/(10−3𝑝)}𝑢_𝜌^𝑝_𝑝, (92) and this means that 𝜌≥6−𝑝6𝜋𝑝^{(10−3𝑝)/8(3−𝑝)}𝑆^3/23𝑝−22𝑝^{(3𝑝−8)/8(3−𝑝)}. (93) Thus(b)holds. At this point, the lemma is proved.

Proof of Theorem 1. It follows from Lemmas 12 and 16 that (8) holds. Let𝑇(𝑢)=∫_ℝ3𝜙_𝑢|𝑢|²𝑑𝑥/4−‖𝑢‖^𝑝_𝑝/𝑝. 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(𝑢_𝜌,𝜆_𝜌). Lemma 17 shows that, for𝑝∈(2,8/3],𝜆_𝜌<0since𝜌₁<𝜌₃, where𝜌₁and𝜌₃are given by (49) and (86), respectively.
To complete the proof of Theorem 1, we need to show that𝜆_𝜌→0and𝐼_𝜌2/𝜌²→0as𝜌→0provided that the assumption(1)of Theorem 1 holds. Indeed, since(𝑢_𝜌,𝜆_𝜌)is the solution of (1), it follows from (87), (88), and Lemma 10 that 30≤4_ℝ³𝜙_𝑢𝜌𝑢_𝜌²𝑑𝑥−𝜆_𝜌𝜌²=6−𝑝𝑢2𝑝^𝜌𝑝𝑝≤𝐶𝜌^{2(6−𝑝)/(10−3𝑝)}, (94) which implies that 0≤−𝜆_𝜌≤𝐶𝜌^{4(𝑝−2)/(10−3𝑝)}, (95) and that is,𝜆_𝜌→0as𝜌→0. On the other hand, we have 𝐼𝑢_𝜌=12∇𝑢_𝜌²₂+14_ℝ³𝜙_𝜌𝑢_𝜌²1𝑑𝑥−𝑝𝑢_𝜌^𝑝_𝑝, (96) this, together with (87) and (88), gives 𝑢_𝜌^𝑝_𝑝=3𝑝2𝐼𝑢𝑝−3^𝜌+𝑝4𝜆3−𝑝^𝜌𝜌². (97) Therefore,𝐼(𝑢_𝜌)<0since𝜆_𝜌<0and𝑝∈(2,8/3). Noting that𝐼(𝑢_𝜌)=𝐼_𝜌2, by Lemma 10 and (97), we obtain 0<3𝑝2𝐼𝑝−3𝜌^𝜌22≤3𝑝−22𝑝^{3(𝑝−2)/(10−3𝑝)}𝑆^{−6(𝑝−2)/(10−3𝑝)}×𝜌^{4(𝑝−2)/(10−3𝑝)}−𝑝4𝜆3−𝑝^𝜌⟶0as𝜌⟶0. (98)

Proof of Theorem 2. Suppose that𝑝∈(2,12/5]and(𝑢_𝜌,𝜆_𝜌)is a solution of (1) with‖𝑢_𝜌‖₂=𝜌. Then Lemma 17 and the above proof of Theorem 1 show that𝜆_𝜌<0,𝐼(𝑢_𝜌)<0and𝜆_𝜌→0as𝜌→0. It was proved in [5, 20] (see also [13, Remark 1.4]) that there exists𝜆₀<0such that (1) has only the zero solution when𝑝∈(2,3)and𝜆∈(−∞,𝜆₀). Therefore,𝜆_𝜌must be bounded; that is,(i)holds. For(ii), it is clear that (87), (88), and (96) hold; after a simple calculation, we have ∇𝑢_𝜌²₂=5𝑝−122𝐼𝑢𝑝−3^𝜌+3𝑝−84𝜆3−𝑝^𝜌𝜌²,_ℝ3𝜙_𝑢𝜌𝑢_𝜌²𝑑𝑥=6−𝑝𝐼𝑢𝑝−3^𝜌+10−3𝑝2𝜆3−𝑝^𝜌𝜌²,𝑢_𝜌^𝑝_𝑝=3𝑝2𝐼𝑢𝑝−3^𝜌+𝑝4𝜆3−𝑝^𝜌𝜌². (99) On the other hand, since𝜙_𝑢𝜌is the solution of the Poisson equation−Δ𝜙=4𝜋|𝑢_𝜌|², multiplying this equation by|𝑢_𝜌|and integrating, we obtain 4𝜋𝑢_𝜌³₃=_ℝ3∇𝜙_𝜌∇𝑢_𝜌≤1𝑑𝑥2∇𝜙_𝜌²₂+12∇𝑢_𝜌²₂=2𝜋_ℝ3𝜙_𝑢𝜌𝑢_𝜌²1𝑑𝑥+2∇𝑢_𝜌²₂. (100) It follows from (99) and (100) that 0<3𝑝2𝐼𝑢𝑝−3^𝜌+𝑝4𝜆3−𝑝^𝜌𝜌²=𝑢_𝜌^𝑝_𝑝≤𝑢_𝜌2^2(3−𝑝)𝑢_𝜌3^3(𝑝−2)≤𝑢_𝜌2^2(3−𝑝)12_ℝ³𝜙_𝑢𝜌𝑢_𝜌²𝑑𝑥+∇𝑢_𝜌²₂^𝑝−2=2𝑝^𝑝−2𝑢_𝜌2^2(3−𝑝)×𝑝2𝑝−32𝐼𝑢𝑝−3^𝜌+𝑝4𝜆3−𝑝^𝜌𝜌^2𝑝−2≤2𝑝^𝑝−2𝑢_𝜌2^2(3−𝑝)×3𝑝2𝐼𝑢𝑝−3^𝜌+𝑝4𝜆3−𝑝^𝜌𝜌^2𝑝−2, (101) which implies that 0<3𝑝2𝐼𝑢𝑝−3^𝜌+𝑝4𝜆3−𝑝^𝜌𝜌²≤2𝑝^{(𝑝−2)/(3−𝑝)}𝑢_𝜌²₂=2𝑝^{(𝑝−2)/(3−𝑝)}𝜌². (102) Therefore we get 0<3𝑝2𝐼𝑢𝑝−3𝜌^𝜌2+𝑝4𝜆3−𝑝^𝜌≤2𝑝^{(𝑝−2)/(3−𝑝)}, (103) so that there exists𝐶>0such that𝐼(𝑢_𝜌)/𝜌²∈(−𝐶,0)since, by(i),𝜆_𝜌<0is 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).