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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 890126, 6 pages
Nonexistence Results for the Schrödinger-Poisson Equations with Spherical and Cylindrical Potentials in
1School of Statistics & Mathematics, Zhongnan University of Economics & Law, Wuhan 430073, China
2Department of Mathematics & Statistics, Curtin University, Perth, WA 6845, Australia
3Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
Received 22 May 2013; Accepted 29 July 2013
Academic Editor: Yonghong Wu
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: , , , where are positive radial functions, , , and is allowed to take two different forms including and with . Two theorems for nonexistence of nontrivial solutions are established, giving two regions on the plane where the system has no nontrivial solutions.
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 : where are positive radial functions, , , and is allowed to have two different forms including and with .
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 [1–4]. For in the form of a constant potential, the nonexistence of nontrivial solutions of (1) for was proved in  by using a Pohožaev-type identity. For 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 plane where the system (1) has no nontrivial solutions. The two 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.
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 , , , and , and for we define By Lemma 2.1 of , we know that has a unique solution in with the form of for any , and By the Hardy-Littlewood-Sobolev inequality, we know that is well defined for any . So we can make the following definition.
Definition 1. For or , if satisfies for all , we say that is a solution of (1).
Now we establish a Pohožaev-type identity based on the work by Badiale et al. . For any , , where , by a simple calculation, we have For any open subset , by using the divergence theorem and (6), we get So, by multiplying (1) by and using (7), we get
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 and , if and , or and , any solution of problem (1) is trivial.
Proof of Theorem 2. Let , , , and ; we then have . Since , , we have So, (9) shows that there exist sequences and such that On we have . By using Cauchy inequality and (10), we get Similarly, we have Hence in (8), by setting , as , from (11) and (12), we have By the second equation of (1), we have From (13) and (14), we get On the other hand, multiplying (1) by and integrating the result over , where , we have Using the divergence theorem to the first term of (16) yields that while the Hölder inequality gives Setting , we have From (16)-(17) and (19), we have By combining (15) and (20), we have For or , we have Then (21) gives that the solution must be trivial.
Theorem 3. For and , if and , or and , any solution of problem (1) with is trivial.
Proof of Theorem 3. For any , setting , then , where and on . Note that Let Then So we must have such that By using Cauchy inequality and (24)–(26), we have It is easy to see that and . Let in (18) by using the definition of and (27), we get Similar to (12), we have such that As , (28)–(29) imply that Since , we have On the other hand, we have So if we multiply (1) by and then integrate over the domain and let , we have As for (20), we have From (31) and (34), we have For or , (35) implies that the solution of problem (1) with , which satisfies , must be trivial.
We mainly study the nonexistence of nontrivial solutions to system (1) in this paper, giving two regions on the 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).
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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