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A nonlinear boundary value problem with many positive solutions. (English) Zbl 0569.35033

The nonlinear problem \(-\Delta u+u=u^{2N+1}\) in G, \(u=0\) on \(\partial G\), where G is the annulus \(\{(x,y):\quad r^ 2<x^ 2+y^ 2<(r+c)^ 2\}\) and N is a positive integer, is considered in this paper. The author proves the following interesting result: for fixed \(c>0\), the number of rotationally non-equivalent positive solutions of this problem tends to infinity as \(r\to +\infty\). As a corollary it is possible to show that for a fixed annulus the number of (rotationally non-equivalent) positive solutions of the problem \(-\Delta u+m^ 2u=u^{2N+1}\) in G, \(u=0\) on \(\partial G\), tends to infinity as \(m\to \infty\). The proof uses a variational principle for a Rayleigh quotient on subspaces of the Sobolev space \(H^ 1_ 0(G)\) with some symmetry properties.
Reviewer: J.Hernández

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
58E30 Variational principles in infinite-dimensional spaces
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