Nonradiality of second fractional eigenfunctions of thin annuli

In the present paper, we study properties of second Dirichlet eigenvalue fractional Laplacian annuli-like domains and corresponding eigenfunctions. first part, consider an annulus with inner radius $ R outer R+1 $. We show that for sufficiently large any eigenfunction this is nonradial. investigate in form B_1(0)\setminus \overline{B_{\tau}(a)} $, where a unitary ball 0<\tau<1-|a| value maximized = 0 if set \overline{B_{\tau}(0)} has no radial eigenfunction. emphasize part our paper implies assumption indeed nonempty.