AI 中文总结
本文研究\u211d⁴中共形不变的指数双调和Choquard方程，完成其共形气泡处线性化算子的完整谱分解，得到尖锐局部稳定性结论，还证明有限质量解为显式平移-伸缩气泡。
AI 中文摘要
我们研究\u211d⁴中共形不变的指数双调和Choquard方程。主要结果是其共形气泡处线性化算子的完整谱分解：经球极投影后，该算子成为Paneitz算子在L²(𝕊⁴)上带紧Riesz分量的有界零阶扰动；我们验证了弱共形转移、消除所有极点支撑的分布缺陷并计算了全部特征值，其Morse指数为1，核空间是五维共形空间，所有高阶模满足一致 coercivity 估计；Adams–Choquard亏格的横向Hessian正是该线性化算子，因此完整谱给出了相对于共形极值流形的Paneitz距离的尖锐局部稳定性；最优渐近常数为γₐ=(160−12α−α²)/(40(10−α))，极限商恰好由二阶球谐函数最小化；作为非线性准备，我们还证明所有正规有限质量分布解满足Niu分类定理的假设，故为显式的平移-伸缩气泡。
英文摘要
We study the conformally invariant exponential biharmonic Choquard equation in \(\mathbb R^{4}\). Our principal result is the complete spectral resolution of the linearized operator at its conformal bubbles. After stereographic projection, the operator becomes a bounded zeroth-order perturbation of the Paneitz operator, with a compact Riesz component on \(L^{2}(\mathbb S^{4})\). We justify the weak conformal transfer, remove every pole-supported distributional defect, and compute all eigenvalues. The Morse index is one. The kernel is the five-dimensional conformal space. All higher modes satisfy a uniform coercivity estimate. The transverse Hessian of the Adams--Choquard deficit is the same linearized operator. Hence the complete spectrum gives sharp local stability with respect to the Paneitz distance from the conformal extremal manifold. The optimal asymptotic constant is \[ γ_α =\frac{160-12α-α^{2}}{40(10-α)}, \] and the limiting quotient is minimized precisely by the second spherical harmonics. As a nonlinear preparation, we also prove that every normal distributional finite-mass solution satisfies the hypotheses of Niu's classification theorem and is therefore an explicit translation--dilation bubble.
