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

Abstract

We consider the elliptic boundary blow-up problem

where $\Omega$ is a smooth bounded domain of ${ℝ}^N$, $a_{+}$, $a_{-}$ 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 $p<N^{*}=\left(N+2\right)∕\left(N-2\right)$. Our proofs are mainly based on continuation methods.