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On a problem of Bieberbach and Rademacher. (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 \[ -\Delta u(x)= p(x) e^{u(x)},\quad x\in \Omega,\quad u_{|\partial\Omega}= \infty, \] where the boundary condition means that \[ \lim_{\begin{smallmatrix} \delta> 0\\ \delta\to 0\end{smallmatrix}} \sup_{\begin{smallmatrix} x\in \Omega\\ d(x, \partial\Omega)< \delta\end{smallmatrix}} u(x)= \infty. \] \(\partial\Omega\) 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 \(p\in C^\alpha(\Omega)\) for some \(\alpha\in (0, 1)\) and if there exists a constant \(k_2\) such that \(0< p(x)\leq k_2\) for all \(x\in \Omega\). If \(p\) is only continuous on \(\Omega\) and satisfies \(0< k_1\leq p(x)\leq k_2\) for all \(x\in \Omega\), then \(|u(x)- \ln(d(x, \partial\Omega)^{- 2}|\) 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 to elliptic equations and elliptic systems
35P15 Estimates of eigenvalues in context of PDEs
35J60 Nonlinear elliptic equations
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