Nodal and asymptotic properties of solutions to nonlinear elliptic eigenvalue problems on general level sets (Q1182835)

The author considers the following nonlinear eigenvalue problem \[ -\Delta u-c(x)u=\lambda f(x,u)\text{ in }\Omega,\quad u=0\text{ on }\partial\Omega, \] where \(\Omega\subset\mathbb{R}^ N\) is a bounded domain with smooth boundary \(\partial\Omega\). The author establishes nodal and asymptotic properties of solutions obtained by Lyusternik-Schnirelman theory on the general level set \[ N_ \alpha:=\{u\in\dot W^{1,2}(\Omega);{1\over 2}\int_ \Omega(|\nabla u|^ 2- c(x)u^ 2)dx=\alpha,\;\alpha<0;\text{ normalizing parameter}\}. \]