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

Jorge Garc�a-Meli�n
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 $$\begin{array}{rlrlr}& \mathrm{\Delta }u=(a_{+}(x)-\epsilon a_{-}(x))u^p& & \text{in }\mathrm{\Omega },& \\ & u=\mathrm{\infty }& & \text{on }\partial \mathrm{\Omega },& \end{array}$$ where $\mathrm{\Omega }$ is a smooth bounded domain of ${\mathbb{R}}^N$, $a_{+}$, $a_{-}$ are positive continuous functions supported in disjoint subdomains ${\mathrm{\Omega }}_{+}$, ${\mathrm{\Omega }}_{-}$ of $\mathrm{\Omega }$, respectively, $p>1$ and $\epsilon >0$ is a parameter. We show that there exists ${\epsilon }^{\ast }>0$ such that no positive solutions exist when $\epsilon >{\epsilon }^{\ast }$, while a minimal positive solution exists for every $\epsilon \in (0,{\epsilon }^{\ast })$. Under the additional hypotheses that ${\bar{\mathrm{\Omega }}}_{+}$ and ${\bar{\mathrm{\Omega }}}_{-}$ intersect along a smooth $(N-1)$-dimensional manifold $\mathrm{\Gamma }$ and $a_{+}$, $a_{-}$ have a convenient decay near $\mathrm{\Gamma }$, we show that a second positive solution exists for every $\epsilon \in (0,{\epsilon }^{\ast })$ if \(p

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