## 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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### References:

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