This paper investigates the solution theory of a class of prescribed positive Q-curvature equations with power-law singularity at the origin and polynomial growth at infinity in the four-dimensional Euclidean space. We focus on the equation involving the biharmonic operator and an exponential nonlinearity, with the prescribed curvature function combining a singular term and a growth term, where a parameter characterizes the strength of the conical singularity at the origin and another parameter describes the growth rate at infinity. Under the finite total curvature constraint, we systematically analyze the asymptotic behavior of normal solutions, establish the necessary condition for existence, prove the existence and uniqueness of radially symmetric normal solutions, and give a complete characterization of the optimal admissible range of the total curvature. Our main results are as follows: (i) We derive the sharp asymptotic behavior of normal solutions both near the singular origin and at infinity, and establish the Pohozaev identity for the singular Q-curvature equation, which yields a universal necessary condition for the existence of normal solutions. (ii) We prove the existence of radially symmetric normal solutions via the Leray–Schauder fixed point theorem combined with a regularization technique, and establish the uniqueness of radial solutions with respect to the initial value at the origin by the strong maximum principle and monotonicity analysis. (iii) We prove the continuity of the total curvature with respect to the initial value via blow-up analysis and energy quantization, and determine the optimal range of the total curvature: for small growth rates, the necessary and sufficient condition for existence is that the total curvature lies between two critical values; for large growth rates, we give a sharp necessary condition and an explicit sufficient condition for the existence of radial solutions.
Tai et al. (Mon,) studied this question.