2017/08/15 by Guo, Zongming, Huang, Xia, Wang, Liping +1 · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1708.04660
We are interested in the qualitative properties of positive entire solutions u ∈ C4 (ℝn \backslash \0\) of the equation Δ2 u=u(n+4)/(n-4) \mboxin ℝn \backslash \0\ and 0 is a non-removable singularity of u(x). It is known from [Theorem 4.2] that any positive entire solution u of \eqref0.0 is radially symmetric with respect to x=0, i.e. u(x)=u(|x|), and equation \eqref0.0 also admits a special positive entire solution us (x)=((n2 (n-4)2)/(16) )(n-4)/(8) |x|-(n-4)/(2). We first show that u-us changes signs infinitely many times in (0, ∞) for any positive singular entire solution u \not ≡ us in ℝN \backslash \0\ of \eqref0.0. Moreover, equation \eqref0.0 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) t u(et) is 2T-periodic with suitably large T. It is still open that v(t):=e(n-4)/(2) t u(et) is periodic for any positive entire solution u(x) of \eqref0.0.