**SMS scnews item created by Daniel Daners at Thu 26 May 2011 0934**

Type: Seminar

Modified: Thu 26 May 2011 0935

Distribution: World

Expiry: 30 May 2011

**Calendar1: 30 May 2011 1400-1500**

**CalLoc1: Mills Room 202**

Auth: daners@bari.maths.usyd.edu.au

### PDE Seminar

# Elliptic problems with sign-changing weights and boundary blow-up

### Garcia-Melian

Jorge
García-Melián

Universidad de La Laguna, Spain

30 May 2011, 2-3pm, Mills Lecture Room 202 (note the location)

## Abstract

We consider the elliptic boundary blow-up problem
\[ \begin{aligned}
& \Delta u=(a_+(x)-\varepsilon a_-(x)) u^p && \mbox{in } \Omega, &\\
& u=\infty && \mbox{on } \partial\Omega,&
\end{aligned}
\]
where \(\Omega\) is a smooth bounded domain of \(\mathbb R^N\), \(a_+\), \(a_-\)
are positive continuous functions supported in disjoint subdomains
\(\Omega_+\), \(\Omega_-\) of \(\Omega\), respectively, \(p>1\) and
\(\varepsilon>0\) is a parameter. We show that there exists
\(\varepsilon^*>0\) such that no positive solutions exist when
\(\varepsilon>\varepsilon^*\), while a minimal positive solution exists
for every \(\varepsilon\in (0,\varepsilon^*)\). Under the additional
hypotheses that \(\overline \Omega_+\) and \(\overline \Omega_-\) intersect
along a smooth \((N-1)\)-dimensional manifold \(\Gamma\) and \(a_+\), \(a_-\)
have a convenient decay near \(\Gamma\), we show that a second positive
solution exists for every \(\varepsilon\in (0,\varepsilon^*) \) if
\(p

Check also the PDE Seminar page. Enquiries to Florica Cîrstea or Daniel Daners.