On Delaunay solutions of a biharmonic elliptic equation with critical exponent

We are interested in the qualitative properties of positive entire solutions u ε C⁴(R^R\{0}) of the equation (Formula Presented.) and 0 is an non-removable singularity of u(x). It is known from [13, Theorem 4.2] that any positive entire solution u of (0.1) is radially symmetric with respect to x = 0, i.e., u(x) = u(|x|), and equation (0.1) also admits a special positive entire solution (Fomrula presented.). We first show that u - u_s changes signs infinitely many times in (0, ∞) for any positive singular entire solution u ≠= u_s in ℛ^N{0} of (0.1). Moreover, equation (0.1) admits a positive entire singular solution u(x) = u(|x|)) such that the scalar curvature of the conformal metric with conformal factor (Fomrula presented.) is positive and (Fomrula presented.) is 2T-periodic with suitably large T.