Liouville Theorem for the Constant $Q/R$-Curvature Equation

We study positive entire solutions of the flat constant $Q/R$-curvature equation in dimension $n\geq5$. We prove that the weak superharmonicity condition $-Δu\geq0$ already forces every nonconstant solution to be admissible. With this automatic admissibility, we classify all such solutions as positive constants or standard bubbles. We also show that the weak superharmonicity condition is structurally necessary. Once it is removed, there is a one-parameter family of positive radial entire solutions near every positive constant. These solutions converge to a positive constant at infinity, while both $-Δu$ and $-Δ(u^{(n-2)/(n-4)})$ change sign infinitely many times.