Lin, Chang-Shou A classification of solutions of a conformally invariant fourth order equation in \(\mathbb{R}^n\). (English) Zbl 0933.35057 Comment. Math. Helv. 73, No. 2, 206-231 (1998). The author considers the entire solutions of the following conformally invariant equations in \({\mathbb R}^n\): \[ \Delta^2 u=u^{ {n+4} \over {n-4} } \quad \text{for }n\geq 5,\tag{1} \] and \[ \Delta^2 u=6\exp (4u)\quad \text{for }n=4,\tag{2} \] where \(\Delta^2\) denotes the biharmonic operator. He proves that all positive solutions of (1) has the form \(u(x)= C_n\{ \lambda / (1+\lambda^2|x-x_0|^2) \}^{{n-4} \over 2 }\) for some positive constants \(\lambda\), \(C_n\) and for some point \(x_0\in{\mathbb R}^n\) (cf. the result of Gidas, Ni and Nirenberg). He gives a condition such that a solution of (2) has the form \(u(x)= \log \{2\lambda/(1+\lambda^2|x-x_0|^2)\} \). Some other properties are mentioned. Reviewer: N.Nakauchi (Yamaguchi) Cited in 5 ReviewsCited in 272 Documents MSC: 35J60 Nonlinear elliptic equations 35C05 Solutions to PDEs in closed form 35J40 Boundary value problems for higher-order elliptic equations Keywords:elliptic equation; biharmonic operator; scalar curvature PDFBibTeX XMLCite \textit{C.-S. Lin}, Comment. Math. Helv. 73, No. 2, 206--231 (1998; Zbl 0933.35057) Full Text: DOI