A unified asymptotic behavior of boundary blow-up solutions to elliptic equations

Abstract

We study unified asymptotic behavior of boundary blow-up solutions to semilinear elliptic equations of the form $$ \begin{cases} \Delta u=b(x)f(u) , & x\in \Omega ,\\ u(x)=\infty , & x\in\partial\Omega , \end{cases} $$ where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain, $b(x)=0$ on $\partial\Omega$ is a non-negative function on $\Omega$, $f$ is non-negative on $[0,\infty)$, and $f(u)/u$ is increasing on $(0,\infty)$, $f(\mathcal {L}(u))= u^{\rho+r}\mathcal {L}'(u)$ as $u\rightarrow\infty$ with $\rho>1-r$ and $0\leq r\leq1$, where $\mathcal{L}\in C^3([A,\infty))$ satisfying $ \lim_{u\rightarrow\infty}\mathcal{L}(u)=\infty$, $\mathcal {L}'\in NRV_{-r}$. In our previous results, we obtained an explicit unified expression of boundary blow-up solutions when $f$ is normalized regularly varying at infinity with index $\rho/(1-r)>1$ or grows at infinity faster than any power function. The effect of the mean curvature of the boundary in the second-order approximation was also discussed. In this paper, we will establish the second-order approximation of boundary blow-up solutions which depends on the distance of $x$ from the boundary $\partial\Omega$. Our analysis is based on the Karamata regular variation theory.