AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 890126 10.1155/2013/890126 890126 Research Article Nonexistence Results for the Schrödinger-Poisson Equations with Spherical and Cylindrical Potentials in 3 http://orcid.org/0000-0001-5050-9583 Jiang Yongsheng 1 Zhou Yanli 1, 2 Wiwatanapataphee B. 3 http://orcid.org/0000-0002-4850-5134 Ge Xiangyu 1 Wu Yonghong 1 School of Statistics & Mathematics Zhongnan University of Economics & Law Wuhan 430073 China znufe.edu.cn 2 Department of Mathematics & Statistics Curtin University Perth, WA 6845 Australia curtin.edu.au 3 Department of Mathematics Faculty of Science Mahidol University Bangkok 10400 Thailand mahidol.ac.th 2013 10 9 2013 2013 22 05 2013 29 07 2013 2013 Copyright © 2013 Yongsheng Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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:3 are positive radial functions, p(1,+), x=(x1,x2,x3)3, 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.

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 3: (1)-Δu+V(x)u+ϕ(x)u=|u|p-1u,-Δϕ=u2,lim|x|+ϕ=0, where u,ϕ:3 are positive radial functions, x=(x1,x2,x3)3, 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  in the study of an N-body quantum problem, that is, the Hartree-Fock system, Kohn-Sham system and, so forth . For V(x) in the form of a constant potential, the nonexistence of nontrivial solutions of (1) for p(1,5) was proved in  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 . 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. , establish a Pohožaev-type identity. In Section 3, we give two theorems summarizing the nonexistence results we obtained and then prove the theorems.

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)3, D1,2(3)={u(x)L6(3):|u|L2(3)}, r1=(x12+x22+x32)1/2, and r2=(x12+x22)1/2, and for α>0 we define (2)E1={uD1,2(3):31r1αu2dx<},E2={uD1,2(3):31r2αu2dx<;11111u(x)=u(r2,x3){uD1,2(3):31r2αu2dx<;}. By Lemma 2.1 of , we know that -Δϕ(x)=u2 has a unique solution in D1,2(3) with the form of (3)ϕ(x):=ϕu(x)=π43u2(y)|x-y|dy for any uL12/5(3), and (4)ϕu(x)2Cu12/52,3ϕu(x)u2dyCu12/54. By the Hardy-Littlewood-Sobolev inequality, we know that 3ϕu(x)uvdy is well defined for any u,vL2E. So we can make the following definition.

Definition 1.

For i=1 or 2, if (u,ϕ)L2Lp+1EiC2(3{ri=0})×D1,2C2(3{ri=0}) satisfies (5)3(uv+1riαuv)dx+3ϕu(x)uvdx=3|u|p-1uvdx for all vL2Lp+1E, 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. . For any uC2(3{ri=0}), x3{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 Ω3{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-α2Ωu2riαdx,Ω(x·u)|u|pdx=Ω1p+1up+1x·νdσΩ(x·u)|u|pdx=-3p+1Ω|u|p+1dx,Ωϕ(x)u(x·u)dx=Σk=13Ωϕ(x)uukxkdx=12Σk=13Ωϕ(x)(u2)kxkdx=12Σk=13[-Ωu2(xkϕ(x)k)dx-Ωϕ(x)u2dx11111111111+Ωϕu2(xk·νk)dσ]=-12Ωu2(ϕ·x)dx-32Ωϕ(x)u2dx+12Ωϕu2(x·ν)dσ=12ΩΔϕ(ϕ·x)dx-32Ωϕ(x)u2dx+12Ωϕu2(x·ν)dσ=12Ωϕu2(x·ν)dσ+12Ω(x·ϕ)(ϕ·ν)dσ-12Ω|ϕ|2(x·ν)dσ-32Ωϕu2dx+14Ω|ϕ|2dx. So, by multiplying (1) by (x·u) and using (7), we get (8)12Ω|u|2dx+3-α2Ωu2riα-3p+1Ω|u|p+1dx+32Ωϕu2dx-14Ω|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.

For x=(x1,x2,x3)3 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 Theorem <xref ref-type="statement" rid="thm1">2</xref>.

Let >R2>R1>0, BR={x3,|x|<R}, B-R={x3,|x|R}, and Ω=BR2B-R1; we then have Ω=BR1BR2. Since uE1Lp+1, ϕD1,2, we have (9)0drBr|u|2+|ϕ|2+u2|x|α+ϕu2+|u|p+1dσ=3|u|2+|ϕ|2+u2|x|α+ϕu2111111+|u|p+1dx<+. So, (9) shows that there exist sequences R1,nn0 and R2,nn+ such that (10)R1,nBR1,n|u|2+|ϕ|2+u2|x|α+ϕu2+|u|p+1dσn0,R2,nBR2,n|u|2+|ϕ|2+u2|x|α+ϕu2+|u|p+1dσn0. 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,nBR1,n|ϕ|2+|u|2dσn0,|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,nBR1,n12(|u|2-|ϕ|2+u2|x|α+ϕu2)111111111111111+1p+1|u|p+1dσn0. Similarly, we have (12)|BR2,n12(x·ϕ)(ϕ·ν)-(x·u)(u·ν)dσ|R2,nBR2,n|ϕ|2+|u|2dσn0.|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,nBR2,n12(|u|2-|ϕ|2+u2|x|α+ϕu2)111111111111+1p+1|u|p+1dσn0. Hence in (8), by setting Ω=BR2,nB-R1,n, as n, from (11) and (12), we have (13)123|u|2dx+3-α23u2|x|α-3p+13|u|p+1dx+323ϕu2dx-143|ϕ|2dx=0. By the second equation of (1), we have (14)3|ϕ|2dx=3ϕu2dx. From (13) and (14), we get (15)123|u|2dx+3-α23u2|x|α+543ϕu2dx-3p+13|u|p+1dx=0. On the other hand, multiplying (1) by u and integrating the result over Ω, where Ω3{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=Ωuudx-Ωuuνdσ, while the Hölder inequality gives (18)|Ωuuνdσ|{Ωu6dσ}1/6{Ω|u|2dσ}1/2|Ω|1/3. Setting Ω=BR2,nB-R1,n, we have (19)|BR1,nuuνdσ|C{BR1,nu6dσ}1/6{BR1,n|u|2dσ}1/2111111×|R1,n|2/3n0,|BR2,nuuνdσ|C{BR2,nu6dσ}1/6{BR2,n|u|2dσ}1/2111111×|R2,n|2/3n0. From (16)-(17) and (19), we have (20)3|u|2dx+3u2|x|αdx+3ϕu2dx-3|u|p+1dx=0. By combining (15) and (20), we have (21)(12-3p+1)3|u|2dx+(3-α2-3p+1)3u2|x|αdx+(54-3p+1)3ϕu2dx=0. For 1<pmin{(3+α)/(3-α),7/5} or pmax{5,(3+α)/(3-α)}, we have (22)3p+1min{12,3-α2,54}or3p+1max{12,3-α2,54}. Then (21) gives that the solution (u,ϕ)Lp+1(3)E1(3)C2(3{0})×D1,2C2(3) must be trivial.

Let Lloc4(3)={u(x):for  any  open  domain  Ω3,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.

For x=(x1,x2,x3)3 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(3)×Lloc4(3) is trivial.

Proof of Theorem <xref ref-type="statement" rid="thm2">3</xref>.

For any R2>R1>0, setting Ω=ΩR1,R2:={xBR2:r2>R1}, then ΩR1,R2={xBR2:r2R1}{xBR2:r2=R1}:=ΣR1,R2ΓR1,R2, where ΓR1,R2={x3:r2=R1,|x3|<R22-R12} and ν(x)=(-x1/R1,-x2/R1,0) on ΓR1,R2. Note that (23)0R2dR1τR1,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)=R1TR1,R2(+ϕu2+u2r2α+|u|p+1)|u|2+|u|4+|ϕ|2+|ϕ|41111111111111111+ϕu2+u2r2α+|u|p+1)dσ0. Then (25)0R2f(R1)R1dR1<. So we must have R1,nn0 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,nτR1,n,R2(|u|2+|ϕ|2+u2r2α+|u|p+1)dσn0,|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|R2τR1,n,R2(|u|2+|ϕ|2)dσR2[{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,nτR1,n,R2|ϕ|4dσ}1/2111111111111+{R1,nτR1,n,R2|u|4dσ}1/2]n0. It is easy to see that ΣR1,n,R2ΣR1,n+1,R2 and nΣR1,n,R2={xBR2:r20}. Let Ω=ΩR1,R2 in (18) by using the definition of ΩR1,Rn and (27), we get (28)12BR2|u|2dx+3-α2BR2u2r2α-3p+1BR2|u|p+1dx+32BR2ϕu2dx-14BR2|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,nn+ 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,nBR2,n|u|2+|ϕ|2+u2r2α111111111111111+ϕu2+|u|p+1dσn0,|BR2,n12(x·ϕ)(ϕ·ν)-(x·u)(u·ν)dσ|R2,nBR2,n|u|2+|ϕ|2dσn0. As n+, (28)–(29) imply that (30)123|u|2dx+3-α23u2riα-3p+13|u|p+1dx+323ϕu2dx-143|ϕ|2dx=0. Since 3|ϕ|2dx=3ϕu2dx, we have (31)123|u|2dx+3-α23u2r2α+543ϕu2dx-3p+13|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|1/2C·{τR1,n,R2R1,n|u|6dσ}1/6×{τR1,n,R2R1,n|u|3dσ}1/3n0. 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)3|u|2dx+3u2r2αdx+3ϕu2dx-3|u|p+1dx=0. From (31) and (34), we have (35)(12-3p+1)3|u|2dx+(3-α2-3p+1)3u2r2αdx+(54-3p+1)3ϕu2dx=0. For 1<pmin{(3+α)/(3-α),7/5} or pmax{5,(3+α)/(3-α)}, (35) implies that the solution of problem (1) with i=2,(u,ϕ)Lp+1(3)E2(3)C2(3{0})×D1,2C2(3), which satisfies (u,ϕ)Lloc4(3)×Lloc4(3), 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.

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