Weights sharing the same eigenvalue

TL;DR

This paper investigates the spectral properties of a nonlinear differential operator, showing that under certain conditions, the set of solutions to a related eigenvalue problem is closed and not sigma-compact in a Sobolev space.

Contribution

It establishes a new result on the structure of solution sets for a class of nonlinear eigenvalue problems involving weights sharing the same eigenvalue.

Findings

The solution set is closed in the Sobolev space.

The solution set is not sigma-compact.

Conditions on the function f determine the spectral properties.

Abstract

Here is the simplest particular case of our main result: let $f:{\bf R}\to {\bf R}$ be a function of class $C^1$, with $\sup_{\bf R}f'>0$, such that $$\lim_{|\xi|\to +\infty}{{f(\xi)}\over {\xi}}=0\ .$$ Then, for each $\lambda>{{\pi^2}\over {\sup_{\bf R}f'}}$, the set of all $u\in H^1_0(]0,1[)$ for which the problem $$\cases{-v''=\lambda f'(u(x)) v & in $]0,1[$\cr & \cr v(0)=v(1)=0\cr}$$ has a non-zero solution is closed and not $\sigma$-compact in $H^1_0(]0,1[)$.