A classification of solutions of a conformally invariant fourth order equation in $\mathbf{R}^n$

C.-S. Lin

doi:10.1007/s000140050052

JournalscmhVol. 73, No. 2pp. 206–231

A classification of solutions of a conformally invariant fourth order equation in $R^{n}$

C.-S. Lin
National Chung Cheng University, Chia-Yi, Taiwan

$A classification of solutions of a conformally invariant fourth order equation in $\mathbf{R}^n$ cover$

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Abstract

In this paper, we consider the following conformally invariant equations of fourth order

{Δ^{2} u = 6 e^{4 u} e^{4 u} \in L^{1} (R^{4}), in R^{4}, (1)

and

{Δ^{2} u = u^{\frac{n + 4}{n - 4}}, u > 0 in R^{n} for n \geq 5, (2)

where $Δ^{2}$ denotes the biharmonic operator in $R^{n}$ . By employing the method of moving planes, we are able to prove that all positive solutions of (2) are arised from the smooth conformal metrics on $S^{n}$ by the stereograph projection. For equation (1), we prove a necessary and sufficient condition for solutions obtained from the smooth conformal metrics on $S^{4}$ .

Cite this article

C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^{n}$ . Comment. Math. Helv. 73 (1998), no. 2, pp. 206–231

DOI 10.1007/S000140050052