Weights sharing the same eigenvalue

Abstract

Here is the simplest particular case of our main result: let $f:\mathbf{R}\to \mathbf{R}$ be a function of class $C^1$, with ${\sup }_{\mathbf{R}}{f}^{\prime }>0$, such that $$\underset{|\xi |\to +\infty }{\lim }\frac{f(\xi )}{\xi }=0.$$ Then, for each $\lambda >\frac{{\pi }^2}{{\sup }_{\mathbf{R}}{f}^{\prime }}$, the set of all $u\in H_0^1(]0,1[)$ for which the problem $$\cases{-v''=λf'(u(x)) v & in $]0,1[$\cr & \cr v(0)=v(1)=0\cr}$$ has a non-zero solution is closed and not $\sigma$-compact in $H_0^1(]0,1[)$.

References in corpus (1)

• Singular points of non-monotone potential operators

Cited by in corpus (1)

• Singular points of non-monotone potential operators