Korean J. Math.  Vol 23, No 2 (2015)  pp.259-267
DOI: https://doi.org/10.11568/kjm.2015.23.2.259

Dirichlet boundary value problem for a class of the noncooperative elliptic system

Tacksun Jung, Q-Heung Choi


This paper is devoted to investigate the existence of the solutions for a class of the noncooperative elliptic system involving critical Sobolev exponents. We show the existence of the negative solution for the problem. We show the existence of the unique negative solution for the system of the linear part of the problem under some conditions, which is also the negative solution of the nonlinear problem. We also consider the eigenvalue problem of the matrix.


Noncooperative elliptic system, critical Sobolev expo- nents nonlinear term, eigenvalue problem of the matrix.

Subject classification

35J50, 35J55.


†This work was supported by Basic Science Research Program through the Na- tional Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (KRF-2013010343).

Full Text:



A. Ambrosetti and G. Prodi, On the inversion of some differential mappings with singularities between Banach spaces, Ann. Mat. Pura. Appl. 93 (1972), 231–246. (Google Scholar)

K. C. Chang, Ambrosetti-Prodi type results in elliptic systems, Nonlinear Anal- ysis TMA. 51 (2002), 553–566. (Google Scholar)

D. G. de Figueiredo, Lectures on boundary value problems of the Ambrosetti- Prodi type, 12 Semin ́ario Brasileiro de An ́alise, 232–292 (October 1980). (Google Scholar)

D. G. de Figueiredo, On the superlinear Ambrosetti-Prodi Problem, MRC Tech. Rep 2522, May, (1983). (Google Scholar)

D. G. de Figueiredo and Y. Jianfu, Critical superlinear Ambrosetti-Prodi prob- lems, Top. Methods in Nonlinear Analysis, 14 (1) (1999), 59–80. (Google Scholar)

A. M. Micheletti and C. Saccon, Multiple nontrivial solutions for a floating beam equation via critical point theory, J. Differential Equations 170 (2001), 157–179. (Google Scholar)

D. C. de Morais Filho, A Variational approach to an Ambrosetti-Prodi type problem for a system of elliptic equations, Nonlinear Analysis, TMA. 26 (10) (1996), 1655–1668. (Google Scholar)


  • There are currently no refbacks.

ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr