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A remark on the resonance set for a semilinear elliptic equation*

Published online by Cambridge University Press:  14 November 2011

Anna Maria Micheletti
Anna Maria Micheletti
Affiliation:
Istituto di Matematiche Applicate “U.Dini”, Facoltá di Ingegneria, Universitá di Pisa, Via Bonanno, 25B, 56100 Pisa, Italy
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Abstract

We study the resonance set ∑ of pairs (α,β) ∊ ℝ2 for which the problem ∆u + αu+ − βu− = 0 has a nontrivial solution . We show that if λ0, is an eigenvalue of multiplicity two of −Δ, then has measure zero, where are the neighbouring eigenvalues of λ0. Moreover, we have that, if the operator Δ + αIu<0 + βIu < 0 has a kernel of dimension one for(α, β) ∊ ∑ and u ≠ 0 such that Δu + αu+ − βu− = 0, then (α, β) is an isolated point on ∑ ∩ L, where L is the straight line parallel to the diagonal of ℝ+ × ℝ+ through (α, β).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 4 , 1994 , pp. 803 - 809
Copyright
Copyright © Royal Society of Edinburgh 1994

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