*(English)*Zbl 0833.35052

This paper is concerned with the problems of existence, uniqueness and behavior near the boundary for the boundary blow-up problem

where the boundary condition means that

$\partial {\Omega}$ is not necessarily smooth and there is no a priori assumption on the behavior of $u$ near $\partial {\Omega}$. Assume that ${\Omega}$ is a bounded open subset of ${\mathbb{R}}^{N}$ satisfying a uniform external sphere condition. The existence of a solution is proved if $p\in {C}^{\alpha}\left({\Omega}\right)$ for some $\alpha \in (0,1)$ and if there exists a constant ${k}_{2}$ such that $0<p\left(x\right)\le {k}_{2}$ for all $x\in {\Omega}$. If $p$ is only continuous on ${\Omega}$ and satisfies $0<{k}_{1}\le p\left(x\right)\le {k}_{2}$ for all $x\in {\Omega}$, then $|u\left(x\right)-ln(d{(x,\partial {\Omega})}^{-2}|$ is uniformly bounded on ${\Omega}$ and the boundary blow-up problem has at most one solution.

The uniqueness is also proved in a less regular case, when ${\Omega}$ is a bounded star-shaped domain in ${\mathbb{R}}^{N}$ (and when $p$ is continuous and strictly positive on $\overline{{\Omega}}$). The proofs are based on dilatation arguments near the boundary, the comparison with solutions in balls and annuli and the maximum principle. Existence is given by sub- and super-solutions techniques and appropriate regularizations of the domain.

##### MSC:

35J67 | Boundary values of solutions of elliptic equations |

35P15 | Estimation of eigenvalues and upper and lower bounds for PD operators |

35J60 | Nonlinear elliptic equations |