On the number of positive solutions of some weakly nonlinear equations on annular regions. (English) Zbl 0705.35043

We discuss the number of positive solutions of \[ -\Delta u=u^{\alpha}\text{ in } D;\quad u=0\text{ on } \partial D, \] where \(1<\alpha <(m+2)/(m-2)\) \((\alpha >1\) if \(m=2)\) and D is a domain in \(R^ m\). In particular, we prove that, under reasonable hypotheses, the number of positive solutions is unaffected by the introduction of small holes into D. In particular our result implies the uniqueness of the positive solution if D is a true annulus with a small central hole. We also produce a number of counter examples showing how complicated these problems are. We include a general result on the existence of large solutions.
Reviewer: E.N.Dancer


35J65 Nonlinear boundary value problems for linear elliptic equations
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