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A classification of solutions of a conformally invariant fourth order equation in n . (English) Zbl 0933.35057

The author considers the entire solutions of the following conformally invariant equations in n :

Δ 2 u=u n+4 n-4 forn5,(1)


Δ 2 u=6exp(4u)forn=4,(2)

where Δ 2 denotes the biharmonic operator. He proves that all positive solutions of (1) has the form u(x)=C n {λ/(1+λ 2 |x-x 0 | 2 )} n-4 2 for some positive constants λ, C n and for some point x 0 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{2λ/(1+λ 2 |x-x 0 | 2 )}. Some other properties are mentioned.

35J60Nonlinear elliptic equations
35C05Solutions of PDE in closed form
35J40Higher order elliptic equations, boundary value problems