Weights sharing the same eigenvalue

Here is the simplest particular case of our main result: let $f:{\bf R}\to {\bf R}$beafunctionofclass$C^1$,with$\sup_{\bf R}f’>0$,suchthat$${\lim }_{\mathrm{\mid }\xi \mathrm{\mid }\to +\mathrm{\infty }}\frac{f(\xi )}{\xi }=0\mathrm{.}$$Then,foreach$\lambda>{{\pi^2}\over {\sup_{\bf R}f’}}$,thesetofall$u\in H^1_0(]0,1[)$forwhichtheproblem$\cases{-v''=\lambda f'(u(x)) v & in]0,1[$\cr & \cr v(0)=v(1)=0\cr}$$hasanon-zerosolutionisclosedandnot$\sigma$-compactin$H^1_0(]0,1[)$.