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Asymptotic behaviour of nonlinear eigenvalue problems involving \(p\)-Laplacian-type operators. (English) Zbl 1134.35013
The present work is motivated by the study of two asymptotic behaviour problems involving sequences of \(p\)-Laplacian-type operators for which the authors propose a unified approach in the general setting of the convergence of particular critical values of a class of Rayleigh quotients. The first problem deals with the study of the asymptotic behaviour as \(p\to\infty\) of the \(k\)th nonlinear eigenvalue of the \(p\)-Laplacian operator. The main contribution of the authors to this problem is the proof of the convergence for the generalized sequence of the \(k\)th eigenvalues (suitably renormalized) for any positive integer \(k\) and a variational characterization of this limit. The second problem deals with the asymptotic behaviour of \(k\)th eigenvalues associated with a family \((A_\varepsilon)\) of \(p\)-Laplacian operator
\[ A_\varepsilon(v):= -\text{div}(a_\varepsilon(\cdot, \nabla,v(\cdot)) \]
with fixed \(p\). Under suitable assumptions on the data of the problem, the author proves that the limit of any sequence of eigenvalues is an eigenvalue of the limit problem and that the sequence of the first eigenvalues converges to the first eigenvalue of the limit operator.

35B40 Asymptotic behavior of solutions to PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J25 Boundary value problems for second-order elliptic equations
35J60 Nonlinear elliptic equations
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
47J30 Variational methods involving nonlinear operators
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