Nonlinear eigenvalue problems of Schrodinger 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 (*)A(0)y + B(gamma)gamma = Agamma 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 W, B(gamma) is a symmetric operator. The left-hand side is assumed to be related to a certain auxiliary functional psi, and the associated linear problems (**)A(0)v + B(gamma)v = muv are supposed to have non-empty discrete spectrum (gamma is an element of X). We reformulate and generalize the topological method presented by the authors in [10] to construct solutions of (*) on a sphere S-R := {gamma is an element of X \ parallel togammaparallel to(H) = R} whose psi-value is the n-th Ljusternik-Schnirelman level of psi\s(R) and whose corresponding eigenvalue is the n-th eigenvalue of the associated linear problem (**), where R > 0 and n is an element of IN 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.