On Delaunay solutions of a biharmonic elliptic equation with critical exponent

Abstract

We are interested in the qualitative properties of positive entire solutions $$u \in {C^4}({\mathbb{R}^n}\backslash \{ 0\} )$$ of the equation (1)$${\Delta ^2}u = {u^{\frac{{n = 4}}{{n - 4}}}}\,in\,a{^n}\backslash \left\{ 0 \right\}\,{\rm{and}}\,{\rm{0}}\,{\rm}\,{\rm{a}}\,{\rm{non - removable}}\,{\rm{singularity}}\,{\rm{of}}\,{\rm{u(x)}}{\rm{.}}$$ 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 $${u_s}(x) = {\left( {\tfrac{{{n^2}{{(n - 4)}^2}}}{{16}}} \right)^{\tfrac{{n - 4}}{8}}}{\left| x \right|^{ - \tfrac{{n - 4}}{2}}}$$. We first show that u - us changes signs infinitely many times in (0, ∞) for any positive singular entire solution u ≠= us in ℛN{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 $$u\frac{4}{{n - 4}}$$ is positive and $$v\,\left( t \right): = {e^{\frac{{n - 4}}{2}t}}u\left( {{e^t}} \right)$$ is 2T-periodic with suitably large T.