Nonlinear Eigenvalue Problems of Schrödinger Type Admitting Eigenfunctions with Given Spectral Characteristics
Michael Heid, Hans‐Peter Heinz, Tobias Weth · Mathematische Nachrichten · 2002
The following work is an extension of our recent paper [10]. We still deal with nonlinear eigenvalue problems of the form in a real Hilbert space ℋ︁ with a semi-bounded self-adjoint operator A0, while for every y from a dense subspace X of ℋ︁, 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 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 SR ≔ {y ∈ X | ∥y∥ℋ︁ = R} whose ψ-value is the n-th Ljusternik-Schnirelman level of ψ| and 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.