Elliptic equations with competing rapidly varying nonlinearities and boundary blow-up

Abstract

We prove that $\Delta u+au=b(x)f(u)$ possesses a unique positive solution such that $\lim_{{\rm dist}(x,\partial\Omega)\to 0}u(x)=\infty$, where $\Omega$ is a smooth bounded domain in ${{\mathbb R} }^N$ and $a\in {{\mathbb R} }$. Here $b$ is a smooth function on $\overline \Omega$ which is positive in $\Omega$ and may vanish on $\partial{\Omega}$ (possibly at a very degenerate rate such as $\exp(-[{\rm dist}(x,\partial\Omega)]^q)$ with $q < 0$). We assume that $f$ is locally Lipschitz continuous on $[0,\infty)$ with $f(u)/u$ increasing for $u>0$ and $f(u)$ grows at $\infty$ faster than any power $u^p$ ($p>1$). As a distinct feature of this study appears the asymptotic behaviour of the boundary blow-up solution, which breaks up depending on how $b(x)$ vanishes on $\partial\Omega$ and how fast $f$ grows at $\infty$.