On Delaunay Solutions of a Biharmonic Elliptic Equation with Critical Exponent

We are interested in the qualitative properties of positive entire solutions u ∈ C(R\{0}) of the equation (0.1) ∆u = u n+4 n−4 in R\{0} and 0 is a non-removable singularity of u(x). It is known from [Theorem 4.2, [12]] 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 us(x) = ( n(n−4) 16 )n−4 8 |x|− 2 . We first show that u − us changes signs infinitely many times in (0,∞) for any positive singular entire solution u 6≡ us in R\{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 u 4 n−4 is positive and v(t) := e n−4 2 u(e) is 2T -periodic with suitably large T . It is still open that v(t) := e n−4 2 u(e) is periodic for any positive entire solution u(x) of (0.1).