Solutions of fractional Laplacian equations and their Morse indices

In this paper, we study the solutions of the following fractional Laplacian equation{(−Δ)su=f(x,u)inΩ,u=0inΩc. Under some assumptions on the nonlinearity, we will show that the boundedness of solutions is equivalent to the finiteness of their Morse indices. We extend the result of Bahri and Lions (1992) [1], from the Laplacian equations to the fractional Laplacian equations.