Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian

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⊂RnΩ⊂Rn (n⩾2n⩾2) is a bounded smooth domain, s∈(0,1)s∈(0,1), (−Δ)s(−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)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.