Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian

Abstract

This paper is devoted to the existence of nodal and multiple solutions of nonlinear problems involving the fractional Laplacian{(−Δ)su=f(x,u)in Ω,u=0on ∂Ω, where Ω⊂Rn (n⩾2) is a bounded smooth domain, s∈(0,1), (−Δ)s stands for the fractional Laplacian. When f is superlinear and subcritical, we prove the existence of a positive solution, a negative solution and a nodal solution. If f(x,u) is odd in u, we obtain an unbounded sequence of nodal solutions. In addition, the number of nodal domains of the nodal solutions are investigated.