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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