Abstract

We construct multiple sign-changing solutions for the nonhomogeneous nonlocal equation (-Delta(Omega))(s)u = vertical bar u vertical bar(4/N-2s) u + epsilon f(x) in Omega, under zero Dirichlet boundary conditions in a bounded domain Omega in R-N, N > 4s, s is an element of(0, 1], with f is an element of L-infinity (Omega), f >= 0 and f not equal 0. Here, epsilon > 0 is a small parameter, and (-Delta(Omega))(s) represents a type of nonlocal operator sometimes called the spectral fractional Laplacian. We show that the number of sign-changing solutions goes to infinity as epsilon -> 0 when it is assumed that Omega and f have certain smoothness and possess certain symmetries, and we are also able to establish accurately the contribution of the nonhomogeneous term in the found solutions. Our proof relies on the Lyapunov-Schmidt reduction method.