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.


35J67 Boundary values of solutions to elliptic equations and elliptic systems
35P15 Estimates of eigenvalues in context of PDEs
35J60 Nonlinear elliptic equations
Full Text: DOI


[1] Bieberbach, L., δu = EU und die automorphen funktionen, Math. annln, 77, 173-212, (1916)
[2] Rademacher, H., Einige besondere probleme partieller differentialgleichungen, (), 838-845
[3] Keller, J.B., On solutions of δu = f(u), Communs pure appl. math., 10, 503-510, (1957) · Zbl 0090.31801
[4] Loewner, C.; Nirenberg, L., Partial differential equations invariant under conformal or projective transformations, (), 245-272
[5] Pohozaev, S.L.; Pohozaev, S.L., The Dirichlet problem for the equation δu = u2, Dokl. akad. nauk. SSSR, Soviet math., 1, 1143-1146, (1960), English translation · Zbl 0097.08503
[6] Kondrat’ev, V.A.; Nikishkin, V.A.; Kondrat’ev, V.A.; Nikishkin, V.A., Asymptotics, near the boundary, of a solution of a singular boundary-value problem for a semilinear elliptic equation, Differentsial’nye vravneniya, Diff. eqns, 26, 345-348, (1990), English translation · Zbl 0706.35054
[7] Díaz, G.; Letelier, R., Explosive solutions of quasilinear elliptic equations: existence and uniqueness, Nonlinear analysis, 20, 2, 97-125, (1993) · Zbl 0793.35028
[8] Pucci, C., Maximum and minimum first eigenvalues for a class of elliptic operators, Proc. am. math. soc., 17, 788-795, (1966) · Zbl 0149.07601
[9] Smoller, J., Shock-waves and reaction-diffusion equations, (1963), Springer Berlin
[10] Sattinger, D.H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana univ. math. J., 21, 979-1000, (1972) · Zbl 0223.35038
[11] Lazer, A.C.; McKenna, P.J., On a singular nonlinear elliptic boundary value problem, Proc. am. math. soc., 111, 721-730, (1991) · Zbl 0727.35057
[12] Milnor, J., Topology from the differentiable viewpoint, (1965), University of Virginia Press Charlottesville, Virginia · Zbl 0136.20402
[13] Frank, P.; von Mises, R., Die differential- und integralgleichungen der mechanik und physik I, (1943), Rosenberg New York · Zbl 0061.16603
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