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A classification of solutions of a conformally invariant fourth order equation in $$\mathbb{R}^n$$. (English) Zbl 0933.35057
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.

##### 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
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