Deformation of domain and the limit of the variational eigenvalues of semilinear elliptic operators (Q1922843)

The author studies the following semilinear elliptic eigenvalue problem: \[ Lu+f(x,u)=0\quad\text{in }\Omega_r,\quad u=0\quad\text{on }\partial\Omega_r, \] where \(\Omega_r\subset\mathbb{R}^N\) is a bounded domain with a smooth boundary with a positive parameter \(r\) and \(L\) is a second-order uniformly elliptic selfadjoint operator. The main problem is the domain dependency of the set of eigenvalues \(\{\mu_n(r,\alpha)\}^\infty_{n=1}\) defined through the Lyusternik-Schnirelman theory. The author proves under a certain kind of smoothness of the deformation of \(\Omega_r\) that \(\mu_n(r,\alpha)\) varies continuously in \(r\). The condition on \(\Omega_r\) for \(\mu_n(r,\alpha)\) to go to \(\infty\) is also given.