Mountain pass and linking type sign-changing solutions for nonlinear problems involving the fractional Laplacian

In this paper we study the existence of sign-changing solutions for nonlinear problems involving the fractional Laplacian {(-Delta)(s)u-lambda u=f(c, u), x is an element of Omega, u=0, x is an element of R-n \ Omega where Omega subset of R-boolean AND(n >= 2) is a bounded smooth domain, s is an element of(0, 1), (-Delta)(s) denotes the fractional Laplacian, lambda is a real parameter, the nonlinear term f satisfies superlinear and subcritical growth conditions at zero and at infinity. When lambda <= 0, we prove the existence of a positive solution, a negative solution and a sign-changing solution by combing minimax method with invariant sets of descending flow. When lambda >= lambda(s)(1)(where lambda(s)(1) denotes the first eigenvalue of the operator (-Delta)(s) in Omega with homogeneous Dirichlet boundary data), we prove the existence of a sign-changing solution by using a variation of linking type theorems.