## Abstract and Applied Analysis

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

Hyungjin Huh

#### 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 467985, 5 pages.

Dates
First available in Project Euclid: 18 April 2013

https://projecteuclid.org/euclid.aaa/1366306733

Digital Object Identifier
doi:10.1155/2013/467985

Mathematical Reviews number (MathSciNet)
MR3034875

Zentralblatt MATH identifier
1264.81181

#### Citation

Huh, Hyungjin. Nonexistence Results of Semilinear Elliptic Equations Coupled with the Chern-Simons Gauge Field. Abstr. Appl. Anal. 2013 (2013), Article ID 467985, 5 pages. doi:10.1155/2013/467985. https://projecteuclid.org/euclid.aaa/1366306733

#### References

• J. Hong, Y. Kim, and P. Y. Pac, “Multivortex solutions of the abelian Chern-Simons-Higgs theory,” Physical Review Letters, vol. 64, no. 19, pp. 2230–2233, 1990.
• R. Jackiw and E. J. Weinberg, “Self-dual Chern-Simons vortices,” Physical Review Letters, vol. 64, no. 19, pp. 2234–2237, 1990.
• D. Chae and O. Yu. Imanuvilov, “The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory,” Communications in Mathematical Physics, vol. 215, no. 1, pp. 119–142, 2000.
• H. Chan, C.-C. Fu, and C.-S. Lin, “Non-topological multi-vortex solutions to the self-dual Chern-Simons-Higgs equation,” Communications in Mathematical Physics, vol. 231, no. 2, pp. 189–221, 2002.
• K. Choe, “Multiple existence results for the self-dual Chern-Simons-Higgs vortex equation,” Communications in Partial Differential Equations, vol. 34, no. 10–12, pp. 1465–1507, 2009.
• M. Nolasco, “Nontopological $N$-vortex condensates for the self-dual Chern-Simons theory,” Communications on Pure and Applied Mathematics, vol. 56, no. 12, pp. 1752–1780, 2003.
• R. Wang, “The existence of Chern-Simons vortices,” Communications in Mathematical Physics, vol. 137, no. 3, pp. 587–597, 1991.
• R. M. Chen and D. Spirn, “Symmetric Chern-Simons-Higgs vortices,” Communications in Mathematical Physics, vol. 285, no. 3, pp. 1005–1031, 2009.
• J. Han and N. Kim, “Nonself-dual Chern-Simons and Maxwell-Chern-Simons vortices on bounded domains,” Journal of Functional Analysis, vol. 221, no. 1, pp. 167–204, 2005.
• J. Han and N. Kim, “Corrigendum: nonself-dual Chern-Simons and Maxwell-Chern-Simons vortices on bounded domains,” Journal of Functional Analysis, vol. 242, no. 2, p. 674, 2007.
• R. Jackiw and S.-Y. Pi, “Classical and quantal nonrelativistic Chern-Simons theory,” Physical Review, vol. 42, no. 10, pp. 3500–3513, 1990.
• G. V. Dunne, Self-Dual Chern-Simons Theories, 1995.
• P. A. Horvathy and P. Zhang, “Vortices in (abelian) Chern-Simons gauge theory,” Physics Reports, vol. 481, no. 5-6, pp. 83–142, 2009.
• J. Byeon, H. Huh, and J. Seok, “Standing waves of nonlinear Schrödinger equations with the gauge field,” Journal of Functional Analysis, vol. 263, no. 6, pp. 1575–1608, 2012.
• H. Huh, “Standing waves of the Schrodinger equation coupled with the Chern-Simons gauge field,” Journal of Mathematical Physics, vol. 53, no. 6, Article ID 063702, 8 pages, 2012.
• H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations. I. Existence of a ground state,” Archive for Rational Mechanics and Analysis, vol. 82, no. 4, pp. 313–345, 1983.