Opuscula Mathematica
Opuscula Math. 35, no. 1 (), 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.
Cite this article as:
Safa Dridi, Bilel Khamessi, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem in the annulus, Opuscula Math. 35, no. 1 (2015), 21-36, http://dx.doi.org/10.7494/OpMath.2015.35.1.21
 
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ISSN 1232−9274, e-ISSN 2300−6919, DOI https://doi.org/10.7494/OpMath
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