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On the influence of domain shape on the existence of large solutions of some superlinear problems. (English) Zbl 0699.35103

Der Autor untersucht die Existenz großer Lösungen des Problems \[ (*)\quad -\Delta u=u^ p\quad (in\quad \Omega),\quad u=0\quad (auf\quad \partial \Omega) \] über einer beschränkten offenen Menge \(\Omega\) in \({\mathbb{R}}^ m\), wobei \(1<p<(m+2)/(m-2)\) sei. In einer früheren Arbeit [J. Differ. Equations 74, No.1, 120-156 (1988; Zbl 0662.34025)] bewies derselbe Autor, daß \(2^ k-1\) nichttriviale positive Lösungen von kleiner \(L^{\infty}\)-Norm existieren, falls \(\Omega\) Vereinigung von k disjunkten oder sich berührenden Kugeln ist. Die vorliegende sehr interessante und gut geschriebene Arbeit gibt wesentliche Erweiterungen und Präzisierungen; hierbei zeigt sich insbesondere, daß das oben erwähnte Multiplizitätsproblem sehr unstabil von kleinen Verformungen von \(\Omega\) abhängt. Beispielsweise besitzt (*) keine großen (in \(L^{\infty})\) Lösungen, falls \(\Omega\) aus zwei leicht überlappenden Kugeln besteht (sogar wenn die entstehende Kante abgerundet wird), aber (*) besitzt große Lösungen, falls \(\Omega\) eine “Hantel mit schmalem Griff” ist.
Reviewer: J.Appell

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Citations:

Zbl 0662.34025
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References:

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