Abstract

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 = y (*) 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 /, and the associated linear problems A 0 v +B(y)v = v (**) are supposed to have non-empty discrete spectrum (y 2 X).We reformulate and generalize the topological method presented by the authors in [10] to construct solutions of (*) on a sphere SR := fy 2 Xj kyk H = Rg whose /-value is the n-th Ljusternik-Schnirelman level of /j SR and whose corresponding eigenvalue is the n-th eigenvalue of the associated linear problem (**), where R ? 0 and n 2 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.

Citations