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We derive a second order estimate for the first m eigenvalues andeigenfunctions of the linearized Gel'fand problem associated to solutions whichblow-up at m points. This allows us to determine, in some suitable situations,some qualitative properties of the first m eigenfunctions as the number ofpoints of concentration or the multiplicity of the eigenvalue .

On the number of peaks of the eigenfunctions of the linearized Gel'fand problem