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

Abstract

In this note we prove in the nonlinear setting of $CD(K,\infty )$ 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 $C{D}^{\ast }(K,N)$ metric measure spaces with uniformly bounded diameter. Additionally, we show that every element $\lambda$ in the Krasnoselskii spectrum is indeed an eigenvalue, namely there exists a nontrivial $u$ satisfying the eigenvalue equation $-\Delta u=\lambda u$.