Boundary-Blow-Up Problems in a Fractal Domain

Abstract

Assume that $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ with $N\ge 2$, which satisfies a uniform interior and exterior cone condition. We determine uniform a priori lower and upper bounds for the growth of solutions and their gradients, of the problem $\Delta u(x)=f(u(x))(x\in \Omega )$ with boundary blow-up, where $f(t)=e^t$ or $f(t)=t^p$ with $p\in (1,+\infty )$. The boundary estimates imply existence and uniqueness of a solution of the above problem. For $f(t)=t^p$ with $p\in (1,+\infty )$ the solution is positive. These results are used to construct a solution of the problem when $\Omega \subset {\mathbb{R}}^2$ is the von Koch snowflake domain.