Existence and boundary behaviour of solutions for a nonlinear Dirichlet problem in the annulus

Abstract

In this paper, we mainly study the following semilinear Dirichlet problem $ -\Delta u=q(x)f(u),\; u > 0,\;x\in \Omega ,$ $u_{|\partial \Omega }=0,$ where $ \Omega $ is an annulus in $\mathbb{R}^{n},\;\big( n\geq 2\big) .$ The function $f$ is nonnegative in $\mathcal{C}^{1}(0,\infty )$ and $q\in \mathcal{C}_{loc}^{\gamma }(\Omega ),\;(0 < \gamma < 1),$ is positive and satisfies some required hypotheses related to Karamata regular variation theory. We establish the existence of a positive classical solution to this problem. We also give a global boundary behavior of such solution.