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Zbl 0833.35052
Lazer, A.C.; McKenna, P.J.
On a problem of Bieberbach and Rademacher.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 21, No.5, 327-335 (1993). ISSN 0362-546X

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 $$\lim_{\smallmatrix \delta> 0\\ \delta\to 0\endsmallmatrix} \sup_{\smallmatrix x\in \Omega\\ d(x, \partial\Omega)< \delta\endsmallmatrix} u(x)= \infty.$$ $\partial\Omega$ 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 $\bbfR^N$ 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 $0< p(x)\le k_2$ for all $x\in \Omega$. If $p$ is only continuous on $\Omega$ and satisfies $0< k_1\le p(x)\le k_2$ 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.\par The uniqueness is also proved in a less regular case, when $\Omega$ is a bounded star-shaped domain in $\bbfR^N$ (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.
[J.Dolbeault (Paris)]
MSC 2000:
*35J67 Boundary values of solutions of elliptic equations
35P15 Estimation of eigenvalues for PD operators
35J60 Nonlinear elliptic equations

Keywords: boundary blow-up; external sphere condition; sub- and super-solutions; regularizations of the domain

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