Asymptotic behavior of positive large solutions of semilinear Dirichlet problems

Abstract

Let $\Omega $ be a smooth bounded domain in $\mathbb{R}^{n},\ n\geq 2$. This paper deals with the existence and the asymptotic behavior of positive solutions of the following problems \begin{equation*} \Delta u=a(x)u^{\alpha },\alpha >1\text{ and }\Delta u=a(x)e^{u}, \end{equation*} with the boundary condition $u_{\mid \partial \Omega }=+\infty .$ The weight function $a(x)$ is positive in $C_{loc}^{\gamma }(\Omega ),$ $0<\gamma <1$, and satisfies an appropriate assumption related to Karamata regular variation theory. Our arguments are based on the sub-supersolution method.

Keywords

• asymptotic behavior
• positive solution
• sub- and supersolution.