Abstract We study Weyl-type lower bounds for the variational eigenvalues associated with a weighted fractional p -Laplacian on bounded domains with Lipschitz boundary in a critical (limit) case. Our approach is based on a covering technique related to the Besicovitch covering theorem, which allows us to overcome the limitations of standard partition-based arguments in the limit case. As a result, we establish an approximation theorem in fractional Sobolev spaces, and based on this theorem we obtain sharp lower bounds for the variational eigenvalues. In the unweighted case, our results recover the expected Weyl-type behavior and provide an alternative approach to earlier methods. To the best of our knowledge, this is the first application of a covering-based approximation method to nonlinear eigenvalue problems.
Mahir Hasanov (Thu,) studied this question.