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

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

### Garcia-Melian

Jorge García-Melián
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