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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 467985, 5 pages
http://dx.doi.org/10.1155/2013/467985
Research Article

Nonexistence Results of Semilinear Elliptic Equations Coupled with the Chern-Simons Gauge Field

Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea

Received 1 November 2012; Accepted 15 January 2013

Academic Editor: Khalil Ezzinbi

Copyright © 2013 Hyungjin Huh. 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.

Abstract

We discuss the nonexistence of nontrivial solutions for the Chern-Simons-Higgs and Chern-Simons-Schrödinger equations. The Derrick-Pohozaev type identities are derived to prove it.

1. Introduction and Main Results

In this paper, we are concerned with the nonexistence of nontrivial solutions to some elliptic equations coupled with Chern-Simons gauge field. More precisely, let us first consider the following system: which is derived from the system (5) with stationary solution ansatz , , and for . Consider where , , for , is the complex scalar field, is the gauge field, is the covariant derivative for , and denotes the imaginary unit.

The Chern-Simons-Higgs system in (5) was introduced in [1, 2] to deal with the electromagnetic phenomena in planar domain such as fractional quantum Hall effect or high temperature superconductivity. The system in (5) has the conservation of the total energy

The special case with a self-dual potential has received much attention and has been studied by several authors, where one can derive the following system of first-order equations called self-dual equations (see [1, 2]) We note that solutions to the self-dual equations (7) provide solutions to (1)–(4). For the self-dual potential , there are two possible boundary conditions to make the energy finite; either or as . The former boundary condition is called “topological” while the latter “non-topological.” A lot of works have been done for the existence of solutions to the self-dual system [37]. Some existence results for the nonself-dual Chern-Simons-Higgs equations with the topological boundary condtion have been proved in [810]. From the mathematical point of view, it is meaningful to study existence and nonexistence of nontrivial solutions under various conditions on . In this paper, we are concerned with the nonexistence of the non-trivial solution to (1)–(4) with the non-topological boundary condtion. The following is our first result.

Theorem 1. Let be a classical solution of (1)–(4) such that and , for any . Let be a function such that , and , . Assume that where is a constant. Then, one has .

The proof is based on the following Derrick-Pohozaev type identity for (1)–(4): As a typical example, we consider . Then it is easy to check that . If one of the following conditions is satisfied, then we have . (1)For , , we take .(2)For , , we take .(3)For , , we take . Note that for the self-dual potential , we have which is not nonnegative for .

The following Chern-Simons gauged Schrödinger system was proposed in [11] when the second quantized body anyon problem is considered With the stationary solution ansatz , and for , we arrive at

In the special case with the potential , we can derive the following self dual equations [1113] Note that solutions to the self-dual system (16) provide solutions to (12)–(15). The self-dual equations (16) can be transformed into the Liouville equation, an integrable equation whose solutions are explicitly known.

For the nonself-dual potential of the form (, ), some existence and nonexistence results have been studied in [14, 15] under the condition of the radially symmetric solution . We prove the following nonexistence result, under various conditions on , for (12)–(15).

Theorem 2. Let be a classical solution of (1)–(4) such that , and for and . One also assumes that is a function such that and , . (1) If the potential satisfies
then one has .(2)  Suppose that is a real-valued function; that is, and for . If the potential satisfies, for a constant ,
then one has .

The proof is based on the Derrick-Pohozaev type identities (40) and (45) for (12)–(15).

Example 3. For the static solution , we consider the potential . Then, taking , we can check However we have, for the complex solution, which is not nonnegative.
The paper is organized as follows. In Section 2, we prove Theorem 1 by deriving Derrick-Pohozaev type identity. Theorem 2 is proved in Section 3. We conclude this section by giving a few notations. (i) denotes the usual Sobolev space . (ii) and .(iii) the  surface  measure  on  .

2. Proof of Theorem 1

We apply Derrick-Pohozaev argument to derive the identity (9) which prove Theorem 1. From now on, we adopt the summation convention for repeated indices.

Suppose that is a solution of (1)–(4). Multiplying (1) by and integrating over , we obtain

Now we set

Then, integrating by parts and taking real parts, we have

For  , we have where we used the notation and the following identity:

Taking the real parts and integrating by parts, we obtain where we used (2)–(4) in the following way: Combining (23) and (26), we have from the identity (21) Thus we have where is a positive constant. Considering the Sobolev embedding and the condition of Theorem 1, we know that , , , . Applying the idea in [16], we know that there exists a sequence such that and consequently On the other hand, we know from (1) that by taking care of the boundary integral terms as before. Combining (31) and (32), we obtain where is a constant. We are ready to prove Theorem 1.

For , we have . If , then there exists such that for . In the region , we have . Using (4) we have in . On the other hand, from (2) and (3), we deduce that . By (1), we obtain for all . By the condition of and , we conclude that .

For , we have . By the condition of and , we have .

3. Proof of Theorem 2

Repeating the similar argument to the proof of Theorem 1, we derive Derrick-Pohozaev type identities for (12)–(15). Suppose that is a solution of (12)–(15). Multiplying (12) by and integrating over , we obtain

Now we set

Then, integrating by parts and taking real parts, we have

Then we have from the identity (34) Applying the same argument in Section 2, the right hand side of the above equality vanishes. Then we conclude that On the other hand, we know from (12) Combining (38) and (39), we end up with Following the reasoning in Theorem 1, we deduce from the fact .

For the proof of the second result in Theorem 2, we assume . Then (13)–(15) can be rewritten by It is easy to check the following identity: from which we derive, with the condition , for , , Considering , we have from (38) and (39) Then we obtain, for a constant , which proves the second result in Theorem 2.  

Acknowledgments

The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0015866) and also partially supported by the TJ Park Junior Faculty Fellowship.

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