The asymptotic behaviour of the second eigenvalue of the -Laplacian operator as goes to 1 is investigated. The limit setting depends only on the geometry of the domain. In the particular case of a planar disc, it is possible to show that the second eigenfunctions are nonradial if is close enough to 1.
Enea Parini. "The Second Eigenvalue of the -Laplacian as Goes to ." Int. J. Differ. Equ. 2010 (SI3) 1 - 23, 2010. https://doi.org/10.1155/2010/984671