On the eigenvalues of the p&q-fractional Laplacian

We consider the eigenvalue problem for the fractional p & q-Laplacian {(-1)spu+mu(-1)squ+ |u|p-2u +mu |u|q-2u= lambda V(x)|u|p-2u in center dot, u = 0 in D N \center dot, where center dot is an open, bounded, and possibly disconnected domain, lambda is an element of D, 1 < q < p < N/s , <mu> > 0 with a weight function in L infinity (center dot) that is allowed no change of sign. We show that the problem has a continuous spectrum. Moreover, our result reveals a discontinuity property for the spectrum as the parameter mu goes to 0+. In addition, a stability property of eigenvalues as s -> 1- is established.