Continuity of nonlinear eigenvalues in \(\mathrm{CD}(K,\infty )\) spaces with respect to measured Gromov-Hausdorff convergence (Q1751572)

In this note, it is proved in the nonlinear setting of \(\mathrm{CD}(K,\infty )\) spaces the stability of the Krasnoselskii spectrum of the Laplace operator \(-\Delta\) under measured Gromov-Hausdorff convergence, under an additional compactness assumption satisfied, for instance, by sequences of \(\mathrm{CD}^\ast(K,N)\) metric measure spaces with uniformly bounded diameter. Additionally, it is shown that every element \(\lambda\) in the Krasnoselskii spectrum is indeed an eigenvalue, namely there exists a nontrivial \(u\) satisfying the eigenvalue equation \(-\Delta u=\lambda u\).