Nonlinear eigenvalue problems admitting eigenfunctions with known geometric properties (Q1809482)

The authors consider nonlinear eigenvalue problems of the form \[ A_0y+ B(y)y= \lambda y \] in a real Hilbert space \(H\), where \(A_0\) is a semi-bounded self-adjoint operator and, for every \(y\) from a certain dense subspace \(X\) of \(H\), \(B(y)\) is a bounded symmetric linear operator. The left-hand side is assumed to be the gradient of a functional \(\psi\in C^1(X)\), and the associated linear problems \[ A_0 v+ B(y)v= \mu v \] are supposed to have discrete spectrum \((y\in X)\). They present a new topological method that permits, under appropriate assumptions, to construct solutions of the equations on a sphere \(S_R:= \{y\in X:\|y\|_H= R\}\) who's \(\psi\)-valued 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 \(n\in\mathbb{N}\) are given. The authors give in the paper an example to use the method for applications to nonlinear Sturm-Liouville problems, to the nonlinear Hill's equations, to periodic solutions of second-order systems, and elliptic partial differential equations with radial symmetry.