Opuscula Math. 35, no. 1 (2015), 21-36
http://dx.doi.org/10.7494/OpMath.2015.35.1.21

Opuscula Mathematica

# Asymptotic behavior of positive solutions of a semilinear Dirichlet problem in the annulus

Safa Dridi
Bilel Khamessi

Abstract. In this paper, we establish existence and asymptotic behavior of a positive classical solution to the following semilinear boundary value problem: $-\Delta u=q(x)u^{\sigma }\;\text{in}\;\Omega,\quad u_{|\partial\Omega}=0.$ Here $$\Omega$$ is an annulus in $$\mathbb{R}^{n}$$, $$n\geq 3$$, $$\sigma \lt 1$$ and $$q$$ is a positive function in $$\mathcal{C}_{loc}^{\gamma }(\Omega )$$, $$0\lt\gamma \lt 1$$, satisfying some appropriate assumptions related to Karamata regular variation theory. Our arguments combine a method of sub- and supersolutions with Karamata regular variation theory.

Keywords: asymptotic behavior, Dirichlet problem, Karamata function, subsolution, supersolution.

Mathematics Subject Classification: 31C15, 34B27, 35K10.

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1. D. Arcoya, Positive solutions for semilinear Dirichlet problems in an annulus, J. Differential Equations 94 (1991), 217-227.
2. S. Ben Othman, H. Mâagli, S. Masmoudi, M. Zribi, Exact asymptotic behavior near the boundary to the solution for singular nonlinear Dirichlet problems, Nonlinear Anal. 71 (2009), 4137-4150.
3. F. Catrina, Nonexistence of positive radial solutions for a problem with singular potential, Adv. Nonlinear Anal. 3 (2014) 1, 1-13.
4. A.B. Cavalheiro, Existence and uniqueness of the solutions of some degenerate nonlinear elliptic equations, Opuscula Math. 34 (2014) 1, 15-28.
5. R. Chemmam, H. Mâagli, S. Masmoudi, M. Zribi, Combined effects in nonlinear singular elliptic problems in a bounded domain, Adv. Nonlinear Anal. 1 (2012), 301-318.
6. R. Chemmam, A. Dhifli, H. Mâagli, Asymptotic behavior of ground state solutions for semilinear and singular Dirichlet problem, Electron. J. Differential Equations 88 (2011), 1-12.
7. M.G. Crandall, P.H. Rabinowitz, L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), 193-222.
8. X. Garaizar, Existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differential Equations 70 (1987), 69-92.
9. M. Ghergu, V.D. Rădulescu, On a class of sublinear singular elliptic problems with convection term, J. Math. Anal. Appl. 311 (2005), 635-646.
10. M. Ghergu, V.D. Rădulescu, Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Applications, Vol. 37, Oxford University Press, 2008, 320 pp.
11. M. Ghergu, V.D. Rădulescu, Nonlinear PDEs Mathematical Models in Biology, Chemistry and Population Genetics, Springer Monographs in Mathematics, Springer Verlag, 2012.
12. S. Gontara, H. Mâagli, S. Masmoudi, S. Turki, Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem, J. Math. Anal. Appl. 369 (2010), 719-729.
13. S. Jator, Z. Sinkala, Uniqueness of positive radial solutions for $$\Delta u + f(u) = 0$$ in the annulus, Int. J. Pure Appl. Math. 12 (2004), 23-31.
14. H. Mâagli, Asymptotic behavior of positive solutions of a semilinear Dirichlet problems, Nonlinear Anal. 74 (2011), 2941-2947.
15. H. Mâagli, S. Turki, Z. Zine El Abidine, Asymptotic behavior of positive solutions of a semilinear Dirichlet problems outside the unit ball, Electron. J. Differential Equations 95 (2013), 1-14.
16. R. Seneta, Regular Varying Functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin, 1976.
17. M. Tang, Uniqueness of positive radial solutions for $$\Delta u - u + up = 0$$ on an annulus, J. Differential Equations 189 (2003), 148-160.
18. Z. Zhang, The asymptotic behavior of the unique solution for the singular Lane-Emdem-Fowler equation, J. Math. Anal. Appl. 312 (2005), 33-43.
19. H. Wang, On the existence of positive solutions for semilinear elliptic equations in the annulus, J. Differential Equations 109 (1994), 1-7.
• Safa Dridi
• Campus Universitaire, Faculté des Sciences de Tunis, Département de Mathématiques, 2092 Tunis, Tunisia
• Bilel Khamessi
• Campus Universitaire, Faculté des Sciences de Tunis, Département de Mathématiques, 2092 Tunis, Tunisia
• Communicated by Vicentiu D. Radulescu.
• Revised: 2014-05-05.
• Accepted: 2014-05-05.
• Published online: 2014-11-12.