Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry

Abstract

Positive entire solutions of the singular biharmonic equation ∆2u + u-q = 0 in ℝn with q > 1 and n ≥ 3 are considered. We prove that there are infinitely many radial entire solutions with different growth rates close to quadratic. If u(0) is kept fixed we show that a unique minimal entire solution exists, which separates the entire solutions from those with compact support. For the special case n = 3 and q = 7 the function U(r) = √1 /√15 + r2 is the minimal entire solution if u(0) = 15-1/4 is kept fixed.