Nonexistence Results for the Schrödinger-Poisson Equations with Spherical and Cylindrical Potentials in ℝ³

Abstract

We study the following Schrödinger-Poisson system: −Δ𝑢+𝑉(𝑥)𝑢+𝜙𝑢=|𝑢|^𝑝−1𝑢, −Δ𝜙=𝑢², lim_{|𝑥|→+∞}𝜙(𝑥)=0, where 𝑢,𝜙∶ℝ³→ℝ are positive radial functions, 𝑝∈(1,+∞), 𝑥=(𝑥₁,𝑥₂,𝑥₃)∈ℝ³, and 𝑉(𝑥) is allowed to take two different forms including 𝑉(𝑥)=1/(𝑥²₁+𝑥²₂+𝑥²₃)^𝛼/2 and 𝑉(𝑥)=1/(𝑥²₁+𝑥²₂)^𝛼/2 with 𝛼>0. Two theorems for nonexistence of nontrivial solutions are established, giving two regions on the 𝛼−𝑝 plane where the system has no nontrivial solutions.

1. Introduction

Schrödinger-Poisson systems arise in quantum mechanics and have been studied by many researchers in the recent years. A number of researches have been focused on quantum transport in semiconductor devices using both mathematical analysis and numerical analysis. Mathematical analysis plays a very crucial role in any investigation. In this paper, we study the nonexistence of nontrivial solutions for the following system in ℝ³: −Δ𝑢+𝑉(𝑥)𝑢+𝜙(𝑥)𝑢=|𝑢|^𝑝−1𝑢,−Δ𝜙=𝑢²,lim_{|𝑥|→+∞}𝜙=0, (1) where 𝑢,𝜙∶ℝ³→ℝ are positive radial functions, 𝑥=(𝑥₁,𝑥₂,𝑥₃)∈ℝ³, 𝑝∈(1,+∞), and 𝑉(𝑥) is allowed to have two different forms including 𝑉(𝑥)=1/(𝑥²₁+𝑥²₂+𝑥²₃)^𝛼/2 and 𝑉(𝑥)=1/(𝑥²₁+𝑥²₂)^𝛼/2 with 𝛼>0.

The above system was introduced in [1] in the study of an N-body quantum problem, that is, the Hartree-Fock system, Kohn-Sham system and, so forth [1–4]. For 𝑉(𝑥) in the form of a constant potential, the nonexistence of nontrivial solutions of (1) for 𝑝∉(1,5) was proved in [5] by using a Pohožaev-type identity. For 𝑉(𝑥) in the form of the singular potentials as considered in this work, existence of positive solutions has been established under certain assumption [6]. However, the conditions under which nontrivial solutions do not exist have not yet been full established. Hence, in this paper, we study the nonexistence of solutions to the problem (1) with singular potential.

The main contribution of this work is the development of analytical results giving two regions on the 𝛼−𝑝 plane where the system (1) has no nontrivial solutions. The two 𝛼−𝑝 regions are shown in Figure 1. The rest of the paper is organized as follows. In Section 2, we first give some basic definitions and concepts and then, based on the method in Badiale et al. [7], establish a Pohožaev-type identity. In Section 3, we give two theorems summarizing the nonexistence results we obtained and then prove the theorems.

Figure 1 

Diagram showing the two regions on the 𝛼−𝑝 plane where system (1) has no nontrivial solution.

2. Preliminaries and a Pohožaev-Type Identity

Firstly, we briefly introduce some notation and definitions and recall some properties and known results of the second equations (Poisson equation) in (1). Throughout the paper, we let 𝑥=(𝑥₁,𝑥₂,𝑥₃)∈ℝ³, 𝐷^1,2(ℝ³)={𝑢(𝑥)∈𝐿⁶(ℝ³)∶|∇𝑢|∈𝐿²(ℝ³)}, 𝑟₁=(𝑥²₁+𝑥²₂+𝑥²₃)^1/2, and 𝑟₂=(𝑥²₁+𝑥²₂)^1/2, and for 𝛼>0 we define 𝐸₁=𝑢∈𝐷^1,2ℝ³∶_ℝ31𝑟^𝛼₁𝑢²,𝐸𝑑𝑥<∞₂=𝑢∈𝐷^1,2ℝ³∶_ℝ31𝑟^𝛼₂𝑢²𝑢(𝑥)𝑟𝑑𝑥<∞;=𝑢₂,𝑥₃. (2) By Lemma 2.1 of [2], we know that −Δ𝜙(𝑥)=𝑢² has a unique solution in 𝐷^1,2(ℝ³) with the form of 𝜙(𝑥)∶=𝜙_𝑢(𝑥)=𝜋4_ℝ³𝑢²𝑦𝑥−𝑦𝑑𝑦 (3) for any 𝑢∈𝐿^12/5(ℝ³), and ∇𝜙_𝑢(𝑥)₂⩽𝐶‖𝑢‖²_12/5,_ℝ3𝜙_𝑢(𝑥)𝑢²𝑑𝑦⩽𝐶‖𝑢‖⁴_12/5. (4) By the Hardy-Littlewood-Sobolev inequality, we know that ∫_ℝ3𝜙_𝑢(𝑥)𝑢𝑣𝑑𝑦 is well defined for any 𝑢,𝑣∈𝐿²∩𝐸. So we can make the following definition.

Definition 1. For 𝑖=1 or 2, if (𝑢,𝜙)∈𝐿²∩𝐿^𝑝+1∩𝐸_𝑖∩𝐶²(ℝ³⧵{𝑟_𝑖=0})×𝐷^1,2∩𝐶²(ℝ³⧵{𝑟_𝑖=0}) satisfies _ℝ31∇𝑢∇𝑣+𝑟^𝛼_𝑖𝑢𝑣𝑑𝑥+_ℝ3𝜙_𝑢(𝑥)=𝑢𝑣𝑑𝑥_ℝ3|𝑢|^𝑝−1𝑢𝑣𝑑𝑥 (5) for all 𝑣∈𝐿²∩𝐿^𝑝+1∩𝐸, we say that (𝑢,𝜙) is a solution of (1).

Now we establish a Pohožaev-type identity based on the work by Badiale et al. [7]. For any 𝑢∈𝐶²(ℝ³⧵{𝑟_𝑖=0}), 𝑥∈ℝ³⧵{𝑟_𝑖=0}, where 𝑖=1,2, by a simple calculation, we have ())1𝑥⋅∇𝑢Δ𝑢=div(𝑥⋅∇𝑢∇𝑢−2|∇𝑢|²𝑥+12|∇𝑢|²,()𝑢𝑥⋅∇𝑢𝑟^𝛼_𝑖1=div2𝑢²𝑟^𝛼_𝑖𝑥−3−𝛼2𝑢²𝑟^𝛼_𝑖,(𝑥⋅∇𝑢)|𝑢|^𝑝−11𝑢=div|𝑢|𝑝+1^𝑝+1𝑥−3|𝑢|𝑝+1^𝑝+1. (6) For any open subset Ω⊂ℝ³⧵{𝑟_𝑖=0}, by using the divergence theorem and (6), we get _Ω(𝑥⋅∇𝑢)Δ𝑢𝑑𝑥=_𝜕Ω)−1(𝑥⋅∇𝑢)(∇𝑢⋅𝜈2|∇𝑢|²+𝑥⋅𝜈𝑑𝜎_Ω12|∇𝑢|²𝑑𝑥,_Ω()𝑢𝑥⋅∇𝑢𝑟^𝛼_𝑖𝑑𝑥=_𝜕Ω12𝑢²𝑟^𝛼_𝑖𝑥⋅𝜈𝑑𝜎−3−𝛼2_Ω𝑢²𝑟^𝛼_𝑖𝑑𝑥,_Ω(𝑥⋅∇𝑢)|𝑢|^𝑝𝑑𝑥=_𝜕Ω1𝑢𝑝+1^𝑝+1−3𝑥⋅𝜈𝑑𝜎𝑝+1_Ω|𝑢|^𝑝+1𝑑𝑥,_Ω𝜙(𝑥)𝑢(𝑥⋅∇𝑢)𝑑𝑥=Σ³_𝑘=1Ω𝜙(𝑥)𝑢𝑢_𝑘𝑥_𝑘=1𝑑𝑥2Σ³_𝑘=1Ω𝜙(𝑥)𝑢²_𝑘𝑥_𝑘=1𝑑𝑥2Σ³_𝑘=1−_Ω𝑢²𝑥_𝑘𝜙(𝑥)_𝑘𝑑𝑥−_Ω𝜙(𝑥)𝑢²+𝑑𝑥_𝜕Ω𝜙𝑢²𝑥_𝑘⋅𝜈_𝑘1𝑑𝜎=−2_Ω𝑢²3∇𝜙⋅𝑥𝑑𝑥−2_Ω𝜙(𝑥)𝑢²+1𝑑𝑥2_𝜕Ω𝜙𝑢²()=1𝑥⋅𝜈𝑑𝜎2_Ω3Δ𝜙∇𝜙⋅𝑥𝑑𝑥−2_Ω𝜙(𝑥)𝑢²+1𝑑𝑥2_𝜕Ω𝜙𝑢²()=1𝑥⋅𝜈𝑑𝜎2_𝜕Ω𝜙𝑢²()1𝑥⋅𝜈𝑑𝜎+2_𝜕Ω−1𝑥⋅∇𝜙∇𝜙⋅𝜈𝑑𝜎2_𝜕Ω∇𝜙²()−3𝑥⋅𝜈𝑑𝜎2_Ω𝜙𝑢²1𝑑𝑥+4_Ω∇𝜙²𝑑𝑥. (7) So, by multiplying (1) by (𝑥⋅∇𝑢) and using (7), we get 12_Ω|∇𝑢|²𝑑𝑥+3−𝛼2_Ω𝑢²𝑟^𝛼_𝑖−3𝑝+1_Ω|𝑢|^𝑝+1+3𝑑𝑥2_Ω𝜙𝑢²1𝑑𝑥−4_Ω∇𝜙²=𝑑𝑥_𝜕Ω12−()+𝑥⋅∇𝜙∇𝜙⋅𝜈𝑥⋅∇𝑢)(∇𝑢⋅𝜈𝑑𝜎_𝜕Ω12|∇𝑢|²−∇𝜙²+𝑢²𝑟^𝛼_𝑖+𝜙𝑢²−1|𝑢|𝑝+1^𝑝+1(𝑥⋅𝜈)𝑑𝜎. (8)

3. Nonexistence Results for the System of Pohožaev-Type Identity Equations

The nonexistence results we obtained for system (1) are summarized in the following two theorems.

Theorem 2. For 𝑥=(𝑥₁,𝑥₂,𝑥₃)∈ℝ³ and 𝑉(𝑥)=1/(𝑥²₁+𝑥²₂+𝑥²₃)^𝛼/2, if 𝛼∈(0,3) and 𝑝∈(1,min{7/5,(3+𝛼)/(3−𝛼)})∪[max{5,(3+𝛼)/(3−𝛼)},+∞), or 𝛼∈[3,∞) and 𝑝∈(1,7/5], any solution (𝑢,𝜙) of problem (1) is trivial.

Proof of Theorem 2. Let ∞>𝑅₂>𝑅₁>0, 𝐵_𝑅={𝑥∈ℝ³,|𝑥|<𝑅}, 𝐵_𝑅={𝑥∈ℝ³,|𝑥|⩽𝑅}, and Ω=𝐵_𝑅2⧵𝐵_𝑅1; we then have 𝜕Ω=𝜕𝐵_𝑅1∪𝜕𝐵_𝑅2. Since 𝑢∈𝐸₁∩𝐿^𝑝+1, 𝜙∈𝐷^1,2, we have ^∞₀𝑑𝑟_𝜕𝐵𝑟|∇𝑢|²+∇𝜙²+𝑢²|𝑥|^𝛼+𝜙𝑢²+|𝑢|^𝑝+1=𝑑𝜎_ℝ3|∇𝑢|²+∇𝜙²+𝑢²|𝑥|^𝛼+𝜙𝑢²+|𝑢|^𝑝+1𝑑𝑥<+∞. (9) So, (9) shows that there exist sequences 𝑅^𝑛_1,𝑛→−0 and 𝑅^𝑛_2,𝑛→−+∞ such that 𝑅_{1,𝑛𝜕𝐵𝑅1,𝑛}|∇𝑢|²+∇𝜙²+𝑢²|𝑥|^𝛼+𝜙𝑢²+|𝑢|^𝑝+1𝑑𝜎^𝑛𝑅⟶0,_{2,𝑛𝜕𝐵𝑅2,𝑛}|∇𝑢|²+∇𝜙²+𝑢²|𝑥|^𝛼+𝜙𝑢²+|𝑢|^𝑝+1𝑑𝜎⟶^𝑛0. (10) On 𝜕𝐵_𝑅1,𝑛 we have 𝜈(𝑥)=−𝑥/𝑅_1,𝑛. By using Cauchy inequality and (10), we get _{𝜕𝐵𝑅1,𝑛}12𝑥⋅∇𝜙∇𝜙⋅𝜈−(𝑥⋅∇𝑢)(∇𝑢⋅𝜈)𝑑𝜎⩽𝑅_{1,𝑛𝜕𝐵𝑅1,𝑛}∇𝜙²+|∇𝑢|²𝑑𝜎⟶^𝑛0,_{𝜕𝐵𝑅1,𝑛}12|∇𝑢|²−∇𝜙²+𝑢²|𝑥|^𝛼+𝜙𝑢²−1|𝑢|𝑝+1^𝑝+1×(𝑥⋅𝜈)𝑑𝜎⩽𝑅_{1,𝑛𝜕𝐵𝑅1,𝑛}12|∇𝑢|²−∇𝜙²+𝑢²|𝑥|^𝛼+𝜙𝑢²+1|𝑢|𝑝+1^𝑝+1𝑑𝜎⟶^𝑛0. (11) Similarly, we have _{𝜕𝐵𝑅2,𝑛}12𝑥⋅∇𝜙∇𝜙⋅𝜈−(𝑥⋅∇𝑢)(∇𝑢⋅𝜈)𝑑𝜎⩽𝑅_{2,𝑛𝜕𝐵𝑅2,𝑛}∇𝜙²+|∇𝑢|²𝑑𝜎⟶^𝑛0._{𝜕𝐵𝑅2,𝑛}12|∇𝑢|²−∇𝜙²+𝑢²|𝑥|^𝛼+𝜙𝑢²−1|𝑢|𝑝+1^𝑝+1(𝑥⋅𝜈)𝑑𝜎⩽𝑅_{2,𝑛𝜕𝐵𝑅2,𝑛}12|∇𝑢|²−∇𝜙²+𝑢²|𝑥|^𝛼+𝜙𝑢²+1|𝑢|𝑝+1^𝑝+1𝑑𝜎⟶^𝑛0. (12) Hence in (8), by setting Ω=𝐵_𝑅2,𝑛⧵𝐵_𝑅1,𝑛, as 𝑛→∞, from (11) and (12), we have 12_ℝ³|∇𝑢|²𝑑𝑥+3−𝛼2_ℝ³𝑢²|𝑥|^𝛼−3𝑝+1_ℝ³|𝑢|^𝑝+1+3𝑑𝑥2_ℝ³𝜙𝑢²1𝑑𝑥−4_ℝ³∇𝜙²𝑑𝑥=0. (13) By the second equation of (1), we have _ℝ3∇𝜙²𝑑𝑥=_ℝ3𝜙𝑢²𝑑𝑥. (14) From (13) and (14), we get 12_ℝ³|∇𝑢|²𝑑𝑥+3−𝛼2_ℝ³𝑢²|𝑥|^𝛼+54_ℝ³𝜙𝑢²−3𝑑𝑥𝑝+1_ℝ³|𝑢|^𝑝+1𝑑𝑥=0. (15) On the other hand, multiplying (1) by 𝑢 and integrating the result over Ω, where Ω⊂ℝ³⧵{0}, we have _Ω𝑢−Δ𝑢𝑢+²|𝑥|^𝛼+𝜙(𝑥)𝑢²𝑑𝑥=_Ω|𝑢|^𝑝+1𝑑𝑥. (16) Using the divergence theorem to the first term of (16) yields that _Ω−Δ𝑢𝑢𝑑𝑥=_Ω∇𝑢∇𝑢𝑑𝑥−_𝜕Ω𝑢𝜕𝑢𝜕𝜈𝑑𝜎, (17) while the Hölder inequality gives _𝜕Ω𝑢𝜕𝑢⩽𝜕𝜈𝑑𝜎_𝜕Ω𝑢⁶𝑑𝜎^1/6_𝜕Ω|∇𝑢|²𝑑𝜎^1/2|𝜕Ω|^1/3. (18) Setting Ω=𝐵_𝑅2,𝑛⧵𝐵_𝑅1,𝑛, we have _{𝜕𝐵𝑅1,𝑛}𝑢𝜕𝑢𝜕𝜈𝑑𝜎⩽𝐶_{𝜕𝐵𝑅1,𝑛}𝑢⁶𝑑𝜎^1/6_{𝜕𝐵𝑅1,𝑛}|∇𝑢|²𝑑𝜎^1/2×𝑅_1,𝑛^𝑛2/3⟶0,_{𝜕𝐵𝑅2,𝑛}𝑢𝜕𝑢𝜕𝜈𝑑𝜎⩽𝐶_{𝜕𝐵𝑅2,𝑛}𝑢⁶𝑑𝜎^1/6_{𝜕𝐵𝑅2,𝑛}|∇𝑢|²𝑑𝜎^1/2×𝑅_2,𝑛^𝑛2/3⟶0. (19) From (16)-(17) and (19), we have _ℝ3|∇𝑢|²𝑑𝑥+_ℝ3𝑢²|𝑥|^𝛼𝑑𝑥+_ℝ3𝜙𝑢²−𝑑𝑥_ℝ3|𝑢|^𝑝+1𝑑𝑥=0. (20) By combining (15) and (20), we have 12−3𝑝+1_ℝ³|∇𝑢|²𝑑𝑥+3−𝛼2−3𝑝+1_ℝ³𝑢²|𝑥|^𝛼+5𝑑𝑥4−3𝑝+1_ℝ³𝜙𝑢²𝑑𝑥=0. (21) For 1<𝑝⩽min{(3+𝛼)/(3−𝛼),7/5} or 𝑝⩾max{5,(3+𝛼)/(3−𝛼)}, we have 31𝑝+1⩽min2,3−𝛼2,54or31𝑝+1⩾max2,3−𝛼2,54. (22) Then (21) gives that the solution (𝑢,𝜙)∈𝐿^𝑝+1(ℝ³)∩𝐸₁(ℝ³)∩𝐶²(ℝ³⧵{0})×𝐷^1,2∩𝐶²(ℝ³) must be trivial.

Let 𝐿⁴_loc(ℝ³)={𝑢(𝑥)∶foranyopendomainΩ⊂ℝ³,𝑢(𝑥)∈𝐿⁴(Ω)}. Similar to Theorem 2, we get another nonexistence result to the system (1) with potential function 𝑉(𝑥)=1/(𝑥²₁+𝑥²₂)^𝛼/2.

Theorem 3. For 𝑥=(𝑥₁,𝑥₂,𝑥₃)∈ℝ³ and 𝑉(𝑥)=1/(𝑥²₁+𝑥²₂)^𝛼/2, if 𝛼∈(0,3) and 𝑝∈(1,min{7/5,(3+𝛼)/(3−𝛼)})∪[max{5,(3+𝛼)/(3−𝛼)},+∞), or 𝛼∈[3,∞) and 𝑝∈(1,7/5], any solution (𝑢,𝜙) of problem (1) with (∇𝑢,∇𝜙)∈𝐿⁴_𝑙𝑜𝑐(ℝ³)×𝐿⁴_𝑙𝑜𝑐(ℝ³) is trivial.

Proof of Theorem 3. For any 𝑅₂>𝑅₁>0, setting Ω=Ω_𝑅1,𝑅2∶={𝑥∈𝐵_𝑅2∶𝑟₂>𝑅₁}, then 𝜕Ω_𝑅1,𝑅2={𝑥∈𝜕𝐵_𝑅2∶𝑟₂⩾𝑅₁}∪{𝑥∈𝐵_𝑅2∶𝑟₂=𝑅₁}∶=Σ_𝑅1,𝑅2∪Γ_𝑅1,𝑅2, where Γ_𝑅1,𝑅2={𝑥∈ℝ³∶𝑟₂=𝑅₁,|𝑥₃|<𝑅²₂−𝑅²₁} and 𝜈(𝑥)=(−𝑥₁/𝑅₁,−𝑥₂/𝑅₁,0) on Γ_𝑅1,𝑅2. Note that ^𝑅2₀𝑑𝑅_{1𝜏𝑅12,𝑅}|∇𝑢|²+|∇𝑢|⁴+∇𝜙²+∇𝜙⁴+𝑢²𝑦²+𝜙𝑢²+|𝑢|^𝑝+1=𝑑𝜎_𝐵𝑅2|∇𝑢|²+|∇𝑢|⁴+∇𝜙²+∇𝜙⁴+𝑢²𝑟^𝛼₂+𝜙𝑢²+|𝑢|^𝑝+1𝑑𝑥<∞. (23) Let 𝑓𝑅₁=𝑅_{1𝑇𝑅12,𝑅}|∇𝑢|²+|∇𝑢|⁴+∇𝜙²+∇𝜙⁴+𝜙𝑢²+𝑢²𝑟^𝛼₂+|𝑢|^𝑝+1𝑑𝜎⩾0. (24) Then ^𝑅2₀𝑓𝑅₁𝑅₁𝑑𝑅₁<∞. (25) So we must have 𝑅^𝑛_1,𝑛→−0 such that 𝑓𝑅⟶_1,𝑛^𝑛0. (26) By using Cauchy inequality and (24)–(26), we have _{𝜏𝑅21,𝑛,𝑅}12|∇𝑢|²−∇𝜙²+𝑢²𝑟^𝛼₂+𝜙𝑢²−1|𝑢|𝑝+1^𝑝+1(𝑥⋅𝜈)𝑑𝜎⩽𝑅_{1,𝑛𝜏𝑅21,𝑛,𝑅}|∇𝑢|²+∇𝜙²+𝑢²𝑟^𝛼₂+|𝑢|^𝑝+1𝑑𝜎⟶^𝑛0,_{𝜏𝑅21,𝑛,𝑅}12−(⩽𝑥⋅∇𝜙∇𝜙⋅𝜈𝑥⋅∇𝑢)(∇𝑢⋅𝜈)𝑑𝜎_{𝜏𝑅21,𝑛,𝑅}+𝑥⋅∇𝜙∇𝜙⋅𝜈𝑑𝜎_{𝜏𝑅21,𝑛,𝑅}(𝑥⋅∇𝑢)(∇𝑢⋅𝜈)𝑑𝜎⩽𝑅_{2𝜏𝑅21,𝑛,𝑅}|∇𝑢|²+∇𝜙²𝑑𝜎⩽𝑅_{2𝜏𝑅21,𝑛,𝑅}𝑑𝜎^1/2_{𝜏𝑅21,𝑛,𝑅}∇𝜙⁴𝑑𝜎^1/2+_{𝜏𝑅21,𝑛,𝑅}𝑑𝜎^1/2_{𝜏𝑅21,𝑛,𝑅}|∇𝑢|⁴𝑑𝜎^1/2=√2𝜋𝑅₂^3/2𝑅_{1,𝑛𝜏𝑅21,𝑛,𝑅}∇𝜙⁴𝑑𝜎^1/2+𝑅_{1,𝑛𝜏𝑅21,𝑛,𝑅}|∇𝑢|⁴𝑑𝜎⟶^1/2𝑛0. (27) It is easy to see that Σ_{𝑅1,𝑛,𝑅2}⊂Σ_{𝑅1,𝑛+1,𝑅2} and ∪_𝑛Σ_{𝑅1,𝑛,𝑅2}={𝑥∈𝜕𝐵_𝑅2∶𝑟₂≠0}. Let Ω=Ω_𝑅1,𝑅2 in (18) by using the definition of 𝜕Ω_{𝑅1,𝑅𝑛} and (27), we get 12_𝐵𝑅2|∇𝑢|²𝑑𝑥+3−𝛼2_𝐵𝑅2𝑢²𝑟^𝛼₂−3𝑝+1_𝐵𝑅2|𝑢|^𝑝+1+3𝑑𝑥2_𝐵𝑅2𝜙𝑢²1𝑑𝑥−4_𝐵𝑅2∇𝜙²=𝑑𝑥_{𝜕𝐵𝑅2}12−()+𝑥⋅∇𝜙∇𝜙⋅𝜈𝑥⋅∇𝑢)(∇𝑢⋅𝜈𝑑𝜎_{𝜕𝐵𝑅2}12|∇𝑢|²−∇𝜙²+𝑢²𝑟^𝛼₂+𝜙𝑢²−1|𝑢|𝑝+1^𝑝+1(𝑥⋅𝜈)𝑑𝜎. (28) Similar to (12), we have 𝑅^𝑛_2,𝑛→−+∞ such that _{𝜕𝐵𝑅2,𝑛}12|∇𝑢|²−∇𝜙²+𝑢²𝑟^𝛼₂+𝜙𝑢²−1|𝑢|𝑝+1^𝑝+1(𝑥⋅𝜈)𝑑𝜎⩽𝑅_{2,𝑛𝜕𝐵𝑅2,𝑛}|∇𝑢|²+∇𝜙²+𝑢²𝑟^𝛼₂+𝜙𝑢²+|𝑢|^𝑝+1𝑑𝜎⟶^𝑛0,_{𝜕𝐵𝑅2,𝑛}12𝑥⋅∇𝜙∇𝜙⋅𝜈−(𝑥⋅∇𝑢)(∇𝑢⋅𝜈)𝑑𝜎⩽𝑅_{2,𝑛𝜕𝐵𝑅2,𝑛}|∇𝑢|²+∇𝜙²𝑑𝜎⟶^𝑛0. (29) As 𝑛→+∞, (28)–(29) imply that 12_ℝ³|∇𝑢|²𝑑𝑥+3−𝛼2_ℝ³𝑢²𝑟^𝛼_𝑖−3𝑝+1_ℝ³|𝑢|^𝑝+1+3𝑑𝑥2_ℝ³𝜙𝑢²1𝑑𝑥−4_ℝ³∇𝜙²𝑑𝑥=0. (30) Since ∫_ℝ3|∇𝜙|²𝑑𝑥=∫_ℝ3𝜙𝑢²𝑑𝑥, we have 12_ℝ³|∇𝑢|²𝑑𝑥+3−𝛼2_ℝ³𝑢²𝑟^𝛼₂+54_ℝ³𝜙𝑢²−3𝑑𝑥𝑝+1_ℝ³|𝑢|^𝑝+1𝑑𝑥=0. (31) On the other hand, we have _{𝜏𝑅21,𝑛,𝑅}𝑢()⩽∇𝑢⋅𝜈𝑑𝜎_{𝜏𝑅21,𝑛,𝑅}|𝑢|⁶𝑑𝜎^1/6_{𝜏𝑅21,𝑛,𝑅}|∇𝑢|³𝑑𝜎^1/3𝜏_{𝑅1,𝑛,𝑅2}^1/2≤𝐶⋅_{𝜏𝑅21,𝑛,𝑅}𝑅_1,𝑛|𝑢|⁶𝑑𝜎^1/6×_{𝜏𝑅21,𝑛,𝑅}𝑅_1,𝑛|∇𝑢|³𝑑𝜎^𝑛1/3⟶0. (32) So if we multiply (1) by 𝑢 and then integrate over the domain Ω_{𝑅1,𝑛,𝑅2} and let 𝑛→+∞, we have _𝐵𝑅2|∇𝑢|²𝑑𝑥+_𝐵𝑅2𝑢²𝑟^𝛼₂𝑑𝑥+_𝐵𝑅2𝜙𝑢²𝑑𝑥−_𝐵𝑅2|𝑢|^𝑝+1=𝑑𝑥_{𝜕𝐵𝑅2}𝑢(∇𝑢⋅𝜈)𝑑𝜎. (33) As for (20), we have _ℝ3|∇𝑢|²𝑑𝑥+_ℝ3𝑢²𝑟^𝛼₂𝑑𝑥+_ℝ3𝜙𝑢²𝑑𝑥−_ℝ3|𝑢|^𝑝+1𝑑𝑥=0. (34) From (31) and (34), we have 12−3𝑝+1_ℝ³|∇𝑢|²𝑑𝑥+3−𝛼2−3𝑝+1_ℝ³𝑢²𝑟^𝛼₂+5𝑑𝑥4−3𝑝+1_ℝ³𝜙𝑢²𝑑𝑥=0. (35) For 1<𝑝⩽min{(3+𝛼)/(3−𝛼),7/5} or 𝑝⩾max{5,(3+𝛼)/(3−𝛼)}, (35) implies that the solution of problem (1) with 𝑖=2,(𝑢,𝜙)∈𝐿^𝑝+1(ℝ³)∩𝐸₂(ℝ³)∩𝐶²(ℝ³⧵{0})×𝐷^1,2∩𝐶²(ℝ³), which satisfies (∇𝑢,∇𝜙)∈𝐿⁴_loc(ℝ³)×𝐿⁴_loc(ℝ³), must be trivial.

4. Conclusion

We mainly study the nonexistence of nontrivial solutions to system (1) in this paper, giving two regions on the 𝛼−𝑝 plane where the system (1) has no nontrivial solutions; see Figure 1. In another paper, we will study the existence of nontrivial solutions to system (1).

Acknowledgments

This research was supported by the National Science Foundation of China (NSFC)(11201486), the Chinese National Social Science Foundation (10BJY104) and the Fundamental Research Funds for Central Universities (31541311208). B. Wiwatanapataphee gratefully acknowledges the support of the Faculty of Science, Mahidol University.