# Nonexistence Results for the Schrödinger-Poisson Equations with Spherical and Cylindrical Potentials inℝ3

Authors: Yongsheng Jiang; Yanli Zhou; B. Wiwatanapataphee; Xiangyu Ge
Journal: Abstract and Applied Analysis
 (2013)

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## 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/(x12+x22+x32)α/2 and V(x)=1/(x12+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.


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## Body

## 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 inℝ3:
(1)-Δu+V(x)u+ϕ(x)u=|u|p-1u,-Δϕ=u2,lim|x|→+∞ϕ=0,
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/(x12+x22+x32)α/2 and V(x)=1/(x12+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 theα-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 theα-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, D1,2(ℝ3)={u(x)∈L6(ℝ3):|∇u|∈L2(ℝ3)}, r1=(x12+x22+x32)1/2, and r2=(x12+x22)1/2, and for α>0 we define
(2)E1={u∈D1,2(ℝ3):∫ℝ3‍1r1αu2dx<∞},E2={u∈D1,2(ℝ3):∫ℝ3‍1r2αu2dx<∞;11111u(x)=u(r2,x3){u∈D1,2(ℝ3):∫ℝ3‍1r2αu2dx<∞;}.
By Lemma 2.1 of [2], we know that -Δϕ(x)=u2 has a unique solution in D1,2(ℝ3) with the form of
(3)ϕ(x):=ϕu(x)=π4∫ℝ3‍u2(y)|x-y|dy
for any u∈L12/5(ℝ3), and
(4)∥∇ϕu(x)∥2⩽C∥u∥12/52,∫ℝ3‍ϕu(x)u2dy⩽C∥u∥12/54.
By the Hardy-Littlewood-Sobolev inequality, we know that ∫ℝ3‍ϕu(x)uvdy is well defined for any u,v∈L2∩E. So we can make the following definition.Definition 1.
Fori=1 or 2, if (u,ϕ)∈L2∩Lp+1∩Ei∩C2(ℝ3∖{ri=0})×D1,2∩C2(ℝ3∖{ri=0}) satisfies
(5)∫ℝ3‍(∇u∇v+1riαuv)dx+∫ℝ3‍ϕu(x)uvdx=∫ℝ3‍|u|p-1uvdx
for all v∈L2∩Lp+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∖{ri=0}), x∈ℝ3∖{ri=0}, where i=1,2, by a simple calculation, we have
(6)(x·∇u)Δu=div[(x·∇u)∇u-12|∇u|2x]+12|∇u|2,(x·∇u)uriα=div[12u2riαx]-3-α2u2riα,(x·∇u)|u|p-1u=div[1p+1|u|p+1x]-3p+1|u|p+1.
For any open subset Ω⊂ℝ3∖{ri=0}, by using the divergence theorem and (6), we get
(7)∫Ω‍(x·∇u)Δudx=∫∂Ω‍[(x·∇u)(∇u·ν)-12|∇u|2x·νdσ]∫Ω‍(x·∇u)Δu,dx=+∫Ω‍12|∇u|2dx,∫Ω‍(x·∇u)uriαdx=∫∂Ω‍12u2riαx·νdσ-3-α2∫Ω‍u2riαdx,∫Ω‍(x·∇u)|u|pdx=∫∂Ω‍1p+1up+1x·νdσ∫Ω‍(x·∇u)|u|pdx=-3p+1∫Ω‍|u|p+1dx,∫Ω‍ϕ(x)u(x·∇u)dx=Σk=13∫Ω‍ϕ(x)uukxkdx=12Σk=13∫Ω‍ϕ(x)(u2)kxkdx=12Σk=13[-∫Ω‍u2(xkϕ(x)k)dx-∫Ω‍ϕ(x)u2dx11111111111+∫∂Ω‍ϕu2(xk·νk)dσ]=-12∫Ω‍u2(∇ϕ·x)dx-32∫Ω‍ϕ(x)u2dx+12∫∂Ω‍ϕu2(x·ν)dσ=12∫Ω‍Δϕ(∇ϕ·x)dx-32∫Ω‍ϕ(x)u2dx+12∫∂Ω‍ϕu2(x·ν)dσ=12∫∂Ω‍ϕu2(x·ν)dσ+12∫∂Ω‍(x·∇ϕ)(∇ϕ·ν)dσ-12∫∂Ω‍|∇ϕ|2(x·ν)dσ-32∫Ω‍ϕu2dx+14∫Ω‍|∇ϕ|2dx.
So, by multiplying (1) by (x·∇u) and using (7), we get
(8)12∫Ω‍|∇u|2dx+3-α2∫Ω‍u2riα-3p+1∫Ω‍|u|p+1dx+32∫Ω‍ϕu2dx-14∫Ω‍|∇ϕ|2dx=∫∂Ω‍12(x·∇ϕ)(∇ϕ·ν)-(x·∇u)(∇u·ν)dσ+∫∂Ω‍{12(|∇u|2-|∇ϕ|2+u2riα+ϕu2)11111111111-1p+1|u|p+1}(x·ν)dσ.


## 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.
Forx=(x1,x2,x3)∈ℝ3 and V(x)=1/(x12+x22+x32)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 Theorem2.
Let∞>R2>R1>0, BR={x∈ℝ3,|x|<R}, B-R={x∈ℝ3,|x|⩽R}, and Ω=BR2∖B-R1; we then have ∂Ω=∂BR1∪∂BR2. Since u∈E1∩Lp+1, ϕ∈D1,2, we have
(9)∫0∞‍dr∫∂Br‍|∇u|2+|∇ϕ|2+u2|x|α+ϕu2+|u|p+1dσ=∫ℝ3‍|∇u|2+|∇ϕ|2+u2|x|α+ϕu2111111+|u|p+1dx<+∞.
So, (9) shows that there exist sequences R1,n→n0 and R2,n→n+∞ such that
(10)R1,n∫∂BR1,n‍|∇u|2+|∇ϕ|2+u2|x|α+ϕu2+|u|p+1dσ⟶n0,R2,n∫∂BR2,n‍|∇u|2+|∇ϕ|2+u2|x|α+ϕu2+|u|p+1dσ⟶n0.
On ∂BR1,n we have ν(x)=-x/R1,n. By using Cauchy inequality and (10), we get
(11)|∫∂BR1,n‍12(x·∇ϕ)(∇ϕ·ν)-(x·∇u)(∇u·ν)dσ|⩽R1,n∫∂BR1,n‍|∇ϕ|2+|∇u|2dσ⟶n0,|∫∂BR1,n‍{12(|∇u|2-|∇ϕ|2+u2|x|α+ϕu2)-1p+1|u|p+1}111×(x·ν)dσ|∫∂BR1,n‍{12(|∇u|2-|∇ϕ|2+u2|x|α+ϕu2)-1p+1|u|p+1}|⩽R1,n∫∂BR1,n‍12(|∇u|2-|∇ϕ|2+u2|x|α+ϕu2)111111111111111+1p+1|u|p+1dσ⟶n0.
Similarly, we have
(12)|∫∂BR2,n‍12(x·∇ϕ)(∇ϕ·ν)-(x·∇u)(∇u·ν)dσ|⩽R2,n∫∂BR2,n‍|∇ϕ|2+|∇u|2dσ⟶n0.|∫∂BR2,n‍{12(|∇u|2-|∇ϕ|2+u2|x|α+ϕu2)1111111-1p+1|u|p+1}(x·ν)dσ|∫∂BR2,n‍{12(|∇u|2-|∇ϕ|2+u2|x|α+ϕu2)|⩽R2,n∫∂BR2,n‍12(|∇u|2-|∇ϕ|2+u2|x|α+ϕu2)111111111111+1p+1|u|p+1dσ⟶n0.
Hence in (8), by setting Ω=BR2,n∖B-R1,n, as n→∞, from (11) and (12), we have
(13)12∫ℝ3‍|∇u|2dx+3-α2∫ℝ3‍u2|x|α-3p+1∫ℝ3‍|u|p+1dx+32∫ℝ3‍ϕu2dx-14∫ℝ3‍|∇ϕ|2dx=0.
By the second equation of (1), we have
(14)∫ℝ3‍|∇ϕ|2dx=∫ℝ3‍ϕu2dx.
From (13) and (14), we get
(15)12∫ℝ3‍|∇u|2dx+3-α2∫ℝ3‍u2|x|α+54∫ℝ3‍ϕu2dx-3p+1∫ℝ3‍|u|p+1dx=0.
On the other hand, multiplying (1) by u and integrating the result over Ω, where Ω⊂ℝ3∖{0}, we have
(16)∫Ω‍-Δuu+u2|x|α+ϕ(x)u2dx=∫Ω‍|u|p+1dx.
Using the divergence theorem to the first term of (16) yields that
(17)∫Ω‍-Δuudx=∫Ω‍∇u∇udx-∫∂Ω‍u∂u∂νdσ,
while the Hölder inequality gives
(18)|∫∂Ω‍u∂u∂νdσ|⩽{∫∂Ω‍u6dσ}1/6{∫∂Ω‍|∇u|2dσ}1/2|∂Ω|1/3.
Setting Ω=BR2,n∖B-R1,n, we have
(19)|∫∂BR1,n‍u∂u∂νdσ|⩽C{∫∂BR1,n‍u6dσ}1/6{∫∂BR1,n‍|∇u|2dσ}1/2111111×|R1,n|2/3⟶n0,|∫∂BR2,n‍u∂u∂νdσ|⩽C{∫∂BR2,n‍u6dσ}1/6{∫∂BR2,n‍|∇u|2dσ}1/2111111×|R2,n|2/3⟶n0.
From (16)-(17) and (19), we have
(20)∫ℝ3‍|∇u|2dx+∫ℝ3‍u2|x|αdx+∫ℝ3‍ϕu2dx-∫ℝ3‍|u|p+1dx=0.
By combining (15) and (20), we have
(21)(12-3p+1)∫ℝ3‍|∇u|2dx+(3-α2-3p+1)∫ℝ3‍u2|x|αdx+(54-3p+1)∫ℝ3‍ϕu2dx=0.
For 1<p⩽min{(3+α)/(3-α),7/5} or p⩾max{5,(3+α)/(3-α)}, we have
(22)3p+1⩽min{12,3-α2,54}or3p+1⩾max{12,3-α2,54}.
Then (21) gives that the solution (u,ϕ)∈Lp+1(ℝ3)∩E1(ℝ3)∩C2(ℝ3∖{0})×D1,2∩C2(ℝ3) must be trivial.LetLloc4(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/(x12+x22)α/2.Theorem 3.
Forx=(x1,x2,x3)∈ℝ3 and V(x)=1/(x12+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)∈Lloc4(ℝ3)×Lloc4(ℝ3) is trivial.Proof of Theorem3.
For anyR2>R1>0, setting Ω=ΩR1,R2:={x∈BR2:r2>R1}, then ∂ΩR1,R2={x∈∂BR2:r2⩾R1}∪{x∈BR2:r2=R1}:=ΣR1,R2∪ΓR1,R2, where ΓR1,R2={x∈ℝ3:r2=R1,|x3|<R22-R12} and ν(x)=(-x1/R1,-x2/R1,0) on ΓR1,R2. Note that
(23)∫0R2‍dR1∫τR1,R2‍(+u2|y|2+ϕu2+|u|p+1)|∇u|2+|∇u|4+|∇ϕ|2+|∇ϕ|41111111111111+u2|y|2+ϕu2+|u|p+1)dσ=∫BR2‍(u2r2α+ϕu2+|u|p+1)|∇u|2+|∇u|4+|∇ϕ|2+|∇ϕ|411111111111+u2r2α+ϕu2+|u|p+1)dx<∞.
Let
(24)f(R1)=R1∫TR1,R2‍(+ϕu2+u2r2α+|u|p+1)|∇u|2+|∇u|4+|∇ϕ|2+|∇ϕ|41111111111111111+ϕu2+u2r2α+|u|p+1)dσ⩾0.
Then
(25)∫0R2‍f(R1)R1dR1<∞.
So we must have R1,n→n0 such that
(26)f(R1,n)⟶n0.
By using Cauchy inequality and (24)–(26), we have
(27)|∫τR1,n,R2‍{12(|∇u|2-|∇ϕ|2+u2r2α+ϕu2)11111111-1p+1|u|p+1{12(|∇u|2-|∇ϕ|2+u2r2α+ϕu2)}(x·ν)dσ|⩽R1,n∫τR1,n,R2‍(|∇u|2+|∇ϕ|2+u2r2α+|u|p+1)dσ⟶n0,|∫τR1,n,R2‍[12(x·∇ϕ)(∇ϕ·ν)-(x·∇u)(∇u·ν)]dσ|⩽|∫τR1,n,R2‍(x·∇ϕ)(∇ϕ·ν)dσ|1111+|∫τR1,n,R2‍(x·∇u)(∇u·ν)dσ|⩽R2∫τR1,n,R2‍(|∇u|2+|∇ϕ|2)dσ⩽R2[{∫τR1,n,R2‍dσ}1/2{∫τR1,n,R2‍|∇ϕ|4dσ}1/21111111+{∫τR1,n,R2‍dσ}1/2{∫τR1,n,R2‍|∇u|4dσ}1/2]=2πR23/2[{R1,n∫τR1,n,R2‍|∇ϕ|4dσ}1/2111111111111+{R1,n∫τR1,n,R2‍|∇u|4dσ}1/2]⟶n0.
It is easy to see that ΣR1,n,R2⊂ΣR1,n+1,R2 and ∪nΣR1,n,R2={x∈∂BR2:r2≠0}. Let Ω=ΩR1,R2 in (18) by using the definition of ∂ΩR1,Rn and (27), we get
(28)12∫BR2‍|∇u|2dx+3-α2∫BR2‍u2r2α-3p+1∫BR2‍|u|p+1dx+32∫BR2‍ϕu2dx-14∫BR2‍|∇ϕ|2dx=∫∂BR2‍12(x·∇ϕ)(∇ϕ·ν)-(x·∇u)(∇u·ν)dσ11111+∫∂BR2‍{12(|∇u|2-|∇ϕ|2+u2r2α+ϕu2)1111111111111-1p+1|u|p+1}(x·ν)dσ.
Similar to (12), we have R2,n→n+∞ such that
(29)|∫∂BR2,n‍{12(|∇u|2-|∇ϕ|2+u2r2α+ϕu2)11111111-1p+1|u|p+1}(x·ν)dσ|∫∂BR2,n‍{12(|∇u|2-|∇ϕ|2+u2r2α+ϕu2)|⩽R2,n∫∂BR2,n‍|∇u|2+|∇ϕ|2+u2r2α111111111111111+ϕu2+|u|p+1dσ⟶n0,|∫∂BR2,n‍12(x·∇ϕ)(∇ϕ·ν)-(x·∇u)(∇u·ν)dσ|⩽R2,n∫∂BR2,n‍|∇u|2+|∇ϕ|2dσ⟶n0.
As n→+∞, (28)–(29) imply that
(30)12∫ℝ3‍|∇u|2dx+3-α2∫ℝ3‍u2riα-3p+1∫ℝ3‍|u|p+1dx+32∫ℝ3‍ϕu2dx-14∫ℝ3‍|∇ϕ|2dx=0.
Since ∫ℝ3‍|∇ϕ|2dx=∫ℝ3‍ϕu2dx, we have
(31)12∫ℝ3‍|∇u|2dx+3-α2∫ℝ3‍u2r2α+54∫ℝ3‍ϕu2dx-3p+1∫ℝ3‍|u|p+1dx=0.
On the other hand, we have
(32)|∫τR1,n,R2‍u(∇u·ν)dσ|⩽{∫τR1,n,R2‍|u|6dσ}1/6{∫τR1,n,R2‍|∇u|3dσ}1/3|τR1,n,R2|1/2≤C·{∫τR1,n,R2‍R1,n|u|6dσ}1/6×{∫τR1,n,R2‍R1,n|∇u|3dσ}1/3⟶n0.
So if we multiply (1) by u and then integrate over the domain ΩR1,n,R2 and let n→+∞, we have
(33)∫BR2‍|∇u|2dx+∫BR2‍u2r2αdx+∫BR2‍ϕu2dx-∫BR2‍|u|p+1dx=∫∂BR2‍u(∇u·ν)dσ.
As for (20), we have
(34)∫ℝ3‍|∇u|2dx+∫ℝ3‍u2r2αdx+∫ℝ3‍ϕu2dx-∫ℝ3‍|u|p+1dx=0.
From (31) and (34), we have
(35)(12-3p+1)∫ℝ3‍|∇u|2dx+(3-α2-3p+1)∫ℝ3‍u2r2αdx+(54-3p+1)∫ℝ3‍ϕu2dx=0.
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)∈Lp+1(ℝ3)∩E2(ℝ3)∩C2(ℝ3∖{0})×D1,2∩C2(ℝ3), which satisfies (∇u,∇ϕ)∈Lloc4(ℝ3)×Lloc4(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).



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