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Nonlinear Eigenvalue Problems of Schrödinger Type Admitting Eigenfunctions with Given Spectral Characteristics
Author(s) -
Heid Michael,
Heinz HansPeter,
Weth Tobias
Publication year - 2002
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200207)242:1<91::aid-mana91>3.0.co;2-z
Subject(s) - mathematics , eigenfunction , eigenvalues and eigenvectors , hilbert space , bounded function , nonlinear system , operator (biology) , spectrum (functional analysis) , mathematical analysis , spectral theory , differential operator , pure mathematics , bounded operator , subspace topology , type (biology) , quantum mechanics , biochemistry , physics , chemistry , repressor , transcription factor , gene , ecology , biology
The following work is an extension of our recent paper [10]. We still deal with nonlinear eigenvalue problems of the form$$ A_{0} y + B(y)y = \lambda y \leqno (*) $$in a real Hilbert space ℋ with a semi‐bounded self‐adjoint operator A 0 , while for every y from a dense subspace X of ℋ, B ( y ) is a symmetric operator. The left‐hand side is assumed to be related to a certain auxiliary functional ψ , and the associated linear problems$$ A_{0} v + B(y)v = \mu v \leqno (**) $$are supposed to have non‐empty discrete spectrum ( y ∈ X ). We reformulate and generalize the topological method presented by the authors in [10] to construct solutions of (∗) on a sphere S R ≔ { y ∈ X | ∥ y ∥ ℋ = R } whose ψ ‐value is the n ‐th Ljusternik‐Schnirelman level of ψ | S Rand whose corresponding eigenvalue is the n ‐th eigenvalue of the associated linear problem (∗∗), where R > 0 and n ∈ ℕ are given. In applications, the eigenfunctions thus found share any geometric property enjoyed by an n ‐th eigenfunction of a linear problem of the form (∗∗). We discuss applications to elliptic partial differential equations with radial symmetry.