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

Florica Corina Cîrstea

#### 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 $\infty$.

#### Article information

Source
Adv. Differential Equations, Volume 12, Number 9 (2007), 995-1030.

Dates
First available in Project Euclid: 18 December 2012

Mathematical Reviews number (MathSciNet)
MR2351836

Zentralblatt MATH identifier
1162.35036

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B40: Asymptotic behavior of solutions

#### Citation

Cîrstea, Florica Corina. Elliptic equations with competing rapidly varying nonlinearities and boundary blow-up. Adv. Differential Equations 12 (2007), no. 9, 995--1030. https://projecteuclid.org/euclid.ade/1355867421