Normal conformal metrics on [formula omitted] with singular Q-curvature having power-like growth

Abstract

In this paper, we consider the prescribed Q-curvature equation with a singularityΔ2u=|x|−αK(x)e4uinR4,Λ=∫R4|x|−αK(x)e4udx<+∞. First, we prove that all solutions are radially symmetric for 0<α<4, K>0 constant and Λ=16π2(1−α4). Next, some blow-up results have been studied by using a classification result for K>0 constant and 0<α<4. Finally, we show that equation with K(x)=(1−|x|p) and 0<α<4 has normal solutions (namely solutions which can be written in integral form) if and only if p∈(0,4−α) and8π2(1+p−α4)≤Λ<16π2(1−α4).