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