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An exact multiplicity result for a class of semilinear equations. (English) Zbl 0877.35048
A sharp multiplicity result for positive solutions of some semilinear elliptic problems on balls in \(\mathbb{R}^2\) is proved. The model example is the problem \[ -\Delta u=\lambda u(u-b)(c-u)\quad\text{in }|x|<\mathbb{R},\quad u=0 \quad\text{on }|x|=r, \] where \(0<b<c\), \(c>2b\), and \(\lambda\) is a real parameter. It is easy to see that all solutions are positive. The main result is that there is a \(\lambda_0>0\) such that the problem has exactly two solutions if \(\lambda>\lambda_0\), one if \(\lambda=\lambda_0\) and no solution for \(\lambda_0>\lambda\). The main tool used in the proofs is a local inversion theorem by Crandall-Rabinowitz and the key point in order to obtain this interesting result is to show that eigenfunctions corresponding to the zero eigenvalue of the linearized problem at degenerate critical points do not change sign. It is pointed out that this is the only point where dimension two plays a role. One-dimensional arguments are widely used all along the proofs. Most of the properties of the solution branches, including stability are exhibited as well.

35J65 Nonlinear boundary value problems for linear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B32 Bifurcations in context of PDEs
Full Text: DOI
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