6533b7d8fe1ef96bd126ab9f

RESEARCH PRODUCT

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

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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 ℋ with a semi-bounded self-adjoint operator A0, 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 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∥ℋ = 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.