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 \begin{equation} \label{0.0} Δ^2 u=u^{\frac{n+4}{n-4}} \;\;\mbox{in ${\mathbb{R}}^n\backslash \{0\}$ and 0 is a non-removable singularity of $u(x)$}. \end{equation} It is known from [Theorem 4.2] that any positive entire solution $u$ of \eqref{0.0} is radially symmetric with respect to $x=0$, i.e. $u(x)=u(|x|)$, and equation \eqref{0.0} also admits a special positive entire solution $u_s(x)=(\frac{n^2(n-4{)}^2}{16}{)}^{\frac{n-4}{8}}|x{|}^{-\frac{n-4}{2}}$. We first show that $u-u_s$ changes signs infinitely many times in $(0,\infty )$ for any positive singular entire solution $u\not\equiv u_s$ in ${\mathbb{R}}^N\backslash \{0\}$ of \eqref{0.0}. Moreover, equation \eqref{0.0} 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 $v(t):=e^{\frac{n-4}{2}t}u(e^t)$ is $2T$-periodic with suitably large $T$. It is still open that $v(t):=e^{\frac{n-4}{2}t}u(e^t)$ is periodic for any positive entire solution $u(x)$ of \eqref{0.0}.

21 pages; comments welcome