Nonlinear Eigenvalue Problems of Schrödinger Type Admitting Eigenfunctions with Given Spectral Characteristics

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 S_R ≔ {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.