Uniqueness and asymptotic behavior of boundary blow-up solutions to semilinear elliptic problems with non-standard growth

Abstract

In this article, we analyze uniqueness and asymptotic behavior of boundary blow-up non-negative solutions to the semilinear elliptic equation $$displaylines{ Delta u=b(x)f(u),quad xin Omega,cr u(x)=infty, quad xinpartialOmega, }$$ where $Omegasubsetmathbb{R}^N$ is a bounded smooth domain, b(x) is a non-negative function on $Omega$ and f is non-negative on $[0,infty)$ satisfying some structural conditions. The main novelty of this paper is that uniqueness is established only by imposing a control on their growth on the weights b(x) near $partialOmega$ and the nonlinear term f at infinite, rather than requiring them to have a precise asymptotic behavior. Our proof is based on the method of sub and super-solutions and the Safonov iteration technique.

Keywords

• Boundary blow-up solutions
• uniqueness
• asymptotic behavior