On the Fuik spectrum of non-local elliptic operators

In this article, we study the Fuik spectrum of the fractional Laplace operator which is defined as the set of all such that (-Delta)(s)u = alpha u(+) - beta u(-) in Omega u = 0 in R-n/Omega.} has a non-trivial solution u, where is a bounded domain in with Lipschitz boundary, n > 2s, . The existence of a first nontrivial curve of this spectrum, some properties of this curve , e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to the Fuik spectrum.