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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

-Δu(x)=p(x)e u(x) ,xΩ,u |Ω =,

where the boundary condition means that

lim δ>0δ0 sup xΩd(x,Ω)<δ u(x)=·

Ω is not necessarily smooth and there is no a priori assumption on the behavior of u near Ω. Assume that Ω is a bounded open subset of N satisfying a uniform external sphere condition. The existence of a solution is proved if pC α (Ω) for some α(0,1) and if there exists a constant k 2 such that 0<p(x)k 2 for all xΩ. If p is only continuous on Ω and satisfies 0<k 1 p(x)k 2 for all xΩ, then |u(x)-ln(d(x,Ω) -2 | is uniformly bounded on Ω and the boundary blow-up problem has at most one solution.

The uniqueness is also proved in a less regular case, when Ω is a bounded star-shaped domain in N (and when p is continuous and strictly positive on Ω ¯). 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.

35J67Boundary values of solutions of elliptic equations
35P15Estimation of eigenvalues and upper and lower bounds for PD operators
35J60Nonlinear elliptic equations