Let \(\Omega \subset \mathbb R^n\) be a bounded domain, let \(\{F_i^{(s)}\}_{i=1}^{I(s)}\) be a collection of nonintersecting sets, and let \(\Omega_s:=\Omega \setminus \bigcup_{i=1}^{I(s)}F_i^{(s)}\). The authors consider a sequence of nonlinear eigenvalue problems \[ Lu_s=\lambda_sg({\mathbf x},u_s), {\mathbf x}\in \Omega_s,\quad u_s({\mathbf x})=0, {\mathbf x}\in \partial \Omega_s,\tag{\(1_s\)} \] where \(Lu:=\langle \nabla,{\mathbf f}({\mathbf x},u,\nabla u)\rangle -f_0({\mathbf x},u,\nabla u)\), \({\mathbf f}:\Omega \times \mathbb R\times \mathbb R^n \mapsto \mathbb R^n\), \(f_0:\Omega \times \mathbb R\times \mathbb R^n \mapsto \mathbb R\), \(g:\Omega \times \mathbb R \mapsto \mathbb R\). Under certain conditions they construct a function \(c_0({\mathbf x},u)\) and a limit problem \[ Lu+c_0({\mathbf x},-u)=\lambda g({\mathbf x},u), {\mathbf x}\in \Omega,\quad u({\mathbf x})=0, {\mathbf x}\in \partial \Omega,\tag{2} \] with the following property: there exist ``minimal'' eigenvalues \(\lambda_s\), \(\lambda \) and the corresponding eigenfunctions \(u_s({\mathbf x})\), \(u({\mathbf x})\) of the problems (\(1_s\)) and (2), respectively, such that \(\lim_{s\to \infty }\lambda_s=\lambda \), and \(u_s({\mathbf x})\to u({\mathbf x})\) weakly in \(W_r^1(\Omega)\) for any \(r