PDE Seminar Abstracts

We consider the elliptic boundary blow-up problem

$$\begin{array}{ccccc}\hfill & \Delta u=\left({a}_{+}\left(x\right)-\epsilon {a}_{-}\left(x\right)\right){u}^{p}\hfill & \hfill & \text{in}\Omega ,\hfill & \hfill \\ \hfill & u=\infty \hfill & \hfill & \text{on}\partial \Omega ,\hfill & \hfill \end{array}$$

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 $\epsilon >0$ is a parameter. We show that there exists ${\epsilon}^{*}>0$ such that no positive solutions exist when $\epsilon >{\epsilon}^{*}$, while a minimal positive solution exists for every $\epsilon \in \left(0,{\epsilon}^{*}\right)$. Under the additional hypotheses that ${\overline{\Omega}}_{+}$ and ${\overline{\Omega}}_{-}$ intersect along a smooth $\left(N-1\right)$-dimensional manifold $\Gamma $ and ${a}_{+}$, ${a}_{-}$ have a convenient decay near $\Gamma $, we show that a second positive solution exists for every $\epsilon \in \left(0,{\epsilon}^{*}\right)$ if $p<{N}^{*}=\left(N+2\right)\u2215\left(N-2\right)$. Our proofs are mainly based on continuation methods.

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