PERTURBED FRACTIONAL EIGENVALUE PROBLEMS

Let Omega subset of R-N (N >= 2) be a bounded domain with Lipschitz boundary. For each p is an element of (1, infinity) and s is an element of (0, 1) we denote by (-Delta(p))(s) the fractional (s, p)-Laplacian operator. In this paper we study the existence of nontrivial solutions for a perturbation of the eigenvalue problem (-Delta(p))(s) u = lambda vertical bar u vertical bar(p-2)u, in Omega, u = 0, in R-N \Omega, with a fractional (t, q)-Laplacian operator in the left-hand side of the equation, when t is an element of (0, 1) and q is an element of (1, infinity) are such that s N/p = t - N/q. We show that nontrivial solutions for the perturbed eigenvalue problem exists if and only if parameter lambda is strictly larger than the first eigenvalue of the (s, p) -Laplacian.