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POSITIVE SOLUTIONS FOR ASYMPTOTICALLY LINEAR ELLIPTIC SYSTEMS

Published online by Cambridge University Press:  09 August 2007

CHAOQUAN PENG and
JIANFU YANG
CHAOQUAN PENG
Affiliation:
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O.Box 71010, Wuhan 430071, and Graduate School, Chinese Academy of Sciences, Beijing 100049, The People's Republic of China e-mail: pcq1979@163.com
JIANFU YANG
Affiliation:
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, P.R. China e-mail: jfyang_2000@yahoo.com
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Abstract

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In this paper, we show that the semilinear elliptic systems of the form(0.1)\begin{equation} \label{0.1} \left\{\arraycolsep=1.5pt \begin{array}{@{}lll@{}} -\Delta u+\mu\Delta v=g(x, v), \quad -\Delta v+\lambda\Delta u=f(x, u) & (x\in \Omega),\\[2mm] \quad u=v=0, & (x\in \partial \Omega) \end{array} \right. \end{equation} possess at least one positive solution pair (u, v) ∈ H 1 0(Ω) × H 1 0(Ω), where Ω is a smooth bounded domain in $\mathbb{R}^N$, f(x,t) and g(x, t) are continuous functions on $\Omega\times \mathbb{R}$ and asymptotically linear at infinity.

Keywords

35J6035J65

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 49 , Issue 2 , May 2007, pp. 377 - 390
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

REFERENCES

1. Clement, Ph., Figueiredo, D.G. and Mitidieri, E., Positive solutions of semilinear elliptic systems, Comm PDE., 17 (1992), 923–940.CrossRefGoogle Scholar
2. Costa, D. G. and Miyagaki, O. H., Nontrivial solutions for perturbations of the p-Laplacian on unbounded dimains, J. Math. Anal. Appl. 193 (1995), 737–755.CrossRefGoogle Scholar
3. de Figueiredo, D. G. and Felmer, P. L., On superquadratic elliptic systems, Trans. Amer. Math. Soc. 343 (1994), 99–106.CrossRefGoogle Scholar
4. Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order. (Springer-Verlag, 1977).CrossRefGoogle Scholar
5. Hulshof, J. and de Vorst, R. C. A. M. Van, Differential systems with strongly indefinite linear part, J. Functional Analysis 114 (1993), 32–58.CrossRefGoogle Scholar
6. Kryszewski, W. and Szulkin, A., Generalized linking theorem with an application to semilinear Schrddinger equations, Adv. Diff. Equas. 3 (1998), 441–472.Google Scholar
7. Gongbao, Li and Szulkin, A., An asymptotically periodic Schrddinger equation with indefinite linear part. Comm. Contemp. Math. 4 (2002), 763–776.Google Scholar
8. Gongbao, Li and Jianfu, Yang, Asymptotically linear elliptic system. Comm PDE. 29 (2004), 925–954.Google Scholar
9. Gongbao, Li and Huan-Song, Zhou, The existence of a positive solution to asymptotically linear scalar field equations, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 81–105.Google Scholar
10. Chaoquan, Peng and Jianfu, Yang, Nonnegative solutions for nonlinear elliptic systems, J. Math. Anal. Appl. to appear.Google Scholar
11. Willem, M., Minimax theorems (Birkhauser, 1996).CrossRefGoogle Scholar
12. Zhou, Huan-Song, An application of mountain pass theorem, Acta Math. Sinica, (N. S), 18 (2002), 27–36.CrossRefGoogle Scholar
13. de Vorst, R. C. A. M. Van, Variational identities and applications to differential systems, Arch. Rational Mech. Anal. 116 (1991), 375–398.CrossRefGoogle Scholar