Existence of solutions for the fractional Nirenberg problem with indefinite curvature functions

In this paper, we investigate the following fractional Nirenberg problem: P-sigma(gSn)(u) = c(n,sigma)R-sigma((g) over bar)(x)u(n+2 sigma/n-2 sigma) on S-n, where P-sigma(gSn) is fractional order conformal invariant operator and R-sigma((g) over bar)(x) is the sigma-curvature for (S-n,(g) over bar) with (g) over bar = u(4/n-2 sigma)gS(n) with n >= 2 and sigma is an element of (0, 1). We show the existence results to the above equation employing the variational method and blowing-up analysis method, when the rotationally symmetric and indefinite curvature function R satisfies certain flatness conditions.