Opuscula Math. 36, no. 3 (2016), 315-336
http://dx.doi.org/10.7494/OpMath.2016.36.3.315

Opuscula Mathematica

# Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in a bounded domain

Majda Chaieb
Abdelwaheb Dhifli
Samia Zermani

Abstract. Let $$\Omega$$ be a bounded domain in $$\mathbb{R}^{n}$$ ($$n\geq 2$$) with a smooth boundary $$\partial \Omega$$. We discuss in this paper the existence and the asymptotic behavior of positive solutions of the following semilinear elliptic system \begin{aligned} -\Delta u&=a_{1}(x)u^{\alpha}v^{r}\quad\text{in}\;\Omega ,\;\;\,u|_{\partial\Omega}=0,\\ -\Delta v&=a_{2}(x)v^{\beta}u^{s}\quad\text{in}\;\Omega ,\;\;\,v|_{\partial\Omega }=0.\end{aligned} Here $$r,s\in \mathbb{R}$$, $$\alpha,\beta \lt 1$$ such that $$\gamma :=(1-\alpha)(1-\beta)-rs\gt 0$$ and the functions $$a_{i}$$ ($$i=1,2$$) are nonnegative and satisfy some appropriate conditions with reference to Karamata regular variation theory.

Keywords: semilinear elliptic system, asymptotic behavior, Karamata class, sub-super solution.

Mathematics Subject Classification: 31B25, 34B15, 34B18, 34B27.

Full text (pdf)

1. J. Busca, R. Manàsevich, A Liouville-type theorem for Lane-Emden system, Indiana Univ. Math. J. 51 (2002), 37-51.
2. R. Chemmam, Asymptotic behavior of ground state solutions of some combined nonlinear problems, Mediterr. J. Math. 10 (2013), 1259-1272.
3. R. Chemmam, H. Mâagli, S. Masmoudi, M. Zribi, Combined effects in nonlinear singular elliptic problems in a bounded domain, Adv. Nonlinear Anal. 51 (2012), 301-318.
4. F.-C. Cîrstea, V. Radulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorbtion, C. R. Math. Acad. Sci. Paris 335 (2002) 5, 447-452.
5. F.-C. Cîrstea, V. Radulescu, Boundary blow-up in nonlinear elliptic equations of Bieberbach-Rademacher type, Trans. Amer. Math. Soc. 359 (2007) 7, 3275-3286.
6. Ph. Clément, J. Fleckinger, E. Mitidieri, F. de Thélin, Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Differential Equations 166 (2000), 455-477.
7. R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems, Nonlinear Anal. 39 (2000), 559-568.
8. D.G. de Figueiredo, Semilinear elliptic systems, Proceedings of the Second School on Nonlinear Functional Analysis and Applications to Differential Equations, ICTP Trieste 1997, World Scientific Publishing Company (1998), A. Ambrosetti, K.-C. Chang, I. Ekeland (eds), 122-152.
9. D.G. de Figueiredo, P. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super. Pisa CI. Sci. 21 (1994), 387-397.
10. D.G. de Figueiredo, B. Sirakov, Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems, Math. Ann. 333 (2005), 231-260.
11. V. Ghanmi, H. Mâagli, V. Radulescu, N. Zeddini, Large and bounded solutions for a class of nonlinear Schödinger stationnary systems, Anal. Appl. (Singap.) 7 (2009) 4, 391-404.
12. M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal. 528 (2010), 3295-3318.
13. J. Giacomoni, J. Hernàndez, P. Sauvy, Quasilinear and singular elliptic systems, Adv. in Nonlinear Anal. 2 (2012), 1-42.
14. H. Mâagli, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem, Nonlinear Anal. 74 (2011), 2941-2947.
15. H. Mâagli, S. Ben Othman, R. Chemmam, Asymptotic behavior of positive large solutions of semilinear Dirichlet problems, Electron. J. of Qual. Theory Differ. Equ. 57 (2013), 1-13.
16. H. Mâagli, S. Turki, Z. Zine el Abidine, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem outside the unit ball, Electron. J. Differential Equations 2013 (2013) 95, 1-14.
17. M. Maniwa, Uniqueness and existence of positive solutions for some semilinear elliptic systems, Nonlinear Anal. 59 (2004), 993-999.
18. V. Radulescu, Singular phenomena in nonlinear elliptic problems: From blow-up boundary solutions to equations with singular nonlinearities, [in:] M. Chipot (ed.), Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 4 (2007), pp. 483-591.
19. V. Radulescu, Qualitative Analysis of nonlinear elliptic partial differential equations: monotonicity, analytic and variational methods, Contemporary Mathematics and its Applications, 6. Hindawi Publishing Corporation, New York, 2008.
20. E. Seneta, Regularly Varying Functions, Lectures Notes in Mathematics 508, Springer-Verlag, Berlin-New York, 1976.
21. Z. Zhang, Positive solutions of Lane-Emden systems with negative exponents: Existence, boundary behavior and uniqueness, Nonlinear Anal. 74 (2011), 5544-5553.
• Majda Chaieb
• Faculté des Sciences de Tunis, Département de Mathématiques, Campus Universitaire, 2092 Tunis, Tunisia
• Abdelwaheb Dhifli
• Faculté des Sciences de Tunis, Département de Mathématiques, Campus Universitaire, 2092 Tunis, Tunisia
• Samia Zermani
• Faculté des Sciences de Tunis, Département de Mathématiques, Campus Universitaire, 2092 Tunis, Tunisia
• Communicated by Vicentiu D. Radulescu.
• Received: 2014-06-10.
• Revised: 2015-10-27.
• Accepted: 2015-11-15.
• Published online: 2016-02-21. Cite this article as:
Majda Chaieb, Abdelwaheb Dhifli, Samia Zermani, Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in a bounded domain, Opuscula Math. 36, no. 3 (2016), 315-336, http://dx.doi.org/10.7494/OpMath.2016.36.3.315

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.