New Types of Singular Solutions of Conformal Q-curvature Equations

Abstract

We construct radial and non-radial singular solutions u ∈ C⁴(B∖{0}) with a non-removable singular point x = 0 for the conformal Q-curvature equation

$$\begin{cases}\Delta^{2}u={\rm{e}}^{u}\quad\text{in}\,B\backslash\{0\},\\\int_{B\backslash\{0\}}{\rm{e}}^{u(x)}\,dx<\infty,\end{cases}$$

where B = {x ∈ ℝ⁴: ∣x∣ < 1}. More precisely, we can construct two different types of singular solutions u ∈ C⁴(B∖{0}) of the equation, satisfying

$$\vert x\vert^{2}u(x)\rightarrow - D<0\quad\text{uniformly}\,\text{as}\,\vert x\vert \rightarrow 0$$

for some D₀ ≥ 0 and any D > D₀ ≥ 0, and satisfying

$$u(x)=o(\vert x\vert^{-2})\quad\text{uniformly}\,\text{as}\,\vert x\vert \rightarrow 0$$

Moreover, detailed asymptotic expansions near x = 0 of these radial and non-radial singular solutions can be established. As an application, we can also obtain the existence of two types of solutions \(u\in C^{4}(\mathbb{R}^{4}\backslash\overline{B})\) to the problem

$$\begin{cases}\Delta^{2}u={\rm{e}}^{u}\quad\text{in}\,\mathbb{R}^{4}\backslash\overline{B},\\\int_{\mathbb{R}^{4}\backslash\overline{B}}{\rm{e}}^{u(x)} dx<\infty\end{cases}$$

satisfying |x|⁻²u(x) → − D < 0 uniformly as |x| → ∞ for some D₀ ≥ 0 and any D > D₀, and satisfying u(x) = o(|x|²) uniformly as |x| → ∞.