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

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

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