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
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.
- D. Arcoya, Positive solutions for semilinear Dirichlet problems in an annulus, J. Differential Equations 94 (1991), 217-227.
- 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.
- F. Catrina, Nonexistence of positive radial solutions for a problem with singular potential, Adv. Nonlinear Anal. 3 (2014) 1, 1-13.
- A.B. Cavalheiro, Existence and uniqueness of the solutions of some degenerate nonlinear elliptic equations, Opuscula Math. 34 (2014) 1, 15-28.
- 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.
- 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.
- M.G. Crandall, P.H. Rabinowitz, L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), 193-222.
- X. Garaizar, Existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differential Equations 70 (1987), 69-92.
- M. Ghergu, V.D. Rădulescu, On a class of sublinear singular elliptic problems with convection term, J. Math. Anal. Appl. 311 (2005), 635-646.
- 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.
- M. Ghergu, V.D. Rădulescu, Nonlinear PDEs Mathematical Models in Biology, Chemistry and Population Genetics, Springer Monographs in Mathematics, Springer Verlag, 2012.
- 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.
- 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.
- H. Mâagli, Asymptotic behavior of positive solutions of a semilinear Dirichlet problems, Nonlinear Anal. 74 (2011), 2941-2947.
- 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.
- R. Seneta, Regular Varying Functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin, 1976.
- M. Tang, Uniqueness of positive radial solutions for \(\Delta u - u + up = 0\) on an annulus, J. Differential Equations 189 (2003), 148-160.
- Z. Zhang, The asymptotic behavior of the unique solution for the singular Lane-Emdem-Fowler equation, J. Math. Anal. Appl. 312 (2005), 33-43.
- 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.
- Received: 2014-02-18.
- Revised: 2014-05-05.
- Accepted: 2014-05-05.
- Published online: 2014-11-12.
