Singular solutions to a semilinear biharmonic equation with a general critical nonlinearity

Abstract

We consider positive solutions $u$ of the semilinear biharmonic equation ${\Delta }^{2}u=|x{|}^{-\frac{n+4}{2}}g(|x{|}^{\frac{n-4}{2}}u)$ in ${\mathbb{R}}^n\setminus \{0\}$ with non-removable singularities at the origin. Under natural assumptions on the nonlinearity $g$, we show that $|x{|}^{\frac{n-4}{2}}u$ is a periodic function of $\ln |x|$ and we classify all such solutions.

To V. Maz'ya on the occasion of his 80th birthday; references added