Classification of positive singular solutions to a nonlinear biharmonic equation with critical exponent

Abstract

For n ≥ 5, we consider positive solutions u of the biharmonic equation Δ^2u = u^((n+4)/(n−4)) on R^n∖{0}, with a nonremovable singularity at the origin. We show that ∣∣x∣∣^((n−4)/2)u is a periodic function of ln|x| and we classify all periodic functions obtained in this way. This result is relevant for the description of the asymptotic behavior of local solutions near singularities and for the Q-curvature problem in conformal geometry.