临界指数双调和Choquard方程多泡解的集中性、局部唯一性与Morse指数
Concentration, Local Uniqueness, and Morse Index of Multi-bubble Solutions for a Critical Exponential Biharmonic Choquard Equation
AI总结：
该研究针对临界指数双调和Choquard方程的Navier问题，分析多泡解的集中性，通过泛函ℱ_m的临界点性质得到多泡解的局部唯一性与Morse指数公式，明确了尺度方向的控制项。
AI中文摘要：
设Ω是R⁴中具有C⁶类的有界区域，0<α<4，K是定义在Ω闭包上的C⁴类正函数。我们研究Navier问题：Δ²u=ε^(8−α)K(x)e^(u(x))乘以（积分Ω内K(y)e^(u(y))/|x−y|^α dy），且在∂Ω上满足u=Δu=0。设G为Navier格林函数，H为其正则部分，记M_α=8π²(8−α)。集中点由泛函ℱ_m(ξ)=Σ（i=1到m）[log K(ξ_i)+(M_α/2)H(ξ_i,ξ_i)] + M_αΣ（i<j）G(ξ_i,ξ_j)支配。ℱ_m的每个C¹稳定临界点都会产生正的m泡解，其尺度为ε⁻¹量级，非线性源收敛到M_αΣδ_ξ_i*。若该临界点非退化，则对应的m泡解在固定的缩放调制邻域内（不计置换）局部唯一。线性化算子在H²(Ω)∩H₀¹(Ω)上非退化，且ind(u_ε)=m + ind(-D²ℱ_m(ξ*))。一个临界四维容量控制尺度方向，约化Hessian的膨胀块为正，等于8π²b_α²|logε|⁻¹I_m + o(|logε|⁻¹)，其中b_α=(8−α)/2。
英文摘要：
Let $Ω\subset\mathbb R^4$ be a bounded domain of class $C^6$, let $0<α<4$, and let $K\in C^4(\overlineΩ)$ be positive. We study the Navier problem \[ Δ^2u=\varepsilon^{8-α}K(x)e^{u(x)} \left(\int_Ω\frac{K(y)e^{u(y)}}{|x-y|^α}\,dy\right), \qquad u=Δu=0\quad\text{on }\partialΩ. \] Let $G$ be the Navier Green function, let $H$ be its regular part, and put $M_α=8π^2(8-α)$. The concentration points are governed by \[ \mathcal F_m(\boldsymbolξ) =\sum_{i=1}^m \left[\log K(ξ_i)+\frac{M_α}{2}H(ξ_i,ξ_i)\right] +M_α\sum_{i<j}G(ξ_i,ξ_j). \] Every $C^1$-stable critical point of $\mathcal F_m$ produces a positive $m$-bubble solution whose scales are of order $\varepsilon^{-1}$ and whose nonlinear source converges to $M_α\sum_iδ_{ξ_i^*}$. If the critical point is nondegenerate, the corresponding $m$-bubble solution is locally unique, modulo permutations, in a fixed scaled modulation neighborhood. The linearized operator is nondegenerate on $H^2(Ω)\cap H_0^1(Ω)$, and \[ \operatorname{ind}(u_\varepsilon) =m+\operatorname{ind}\!\left(-D^2\mathcal F_m(\boldsymbolξ^*)\right). \] A critical four-dimensional capacity controls the scale directions. The dilation block of the reduced Hessian is positive and equals $8π^2b_α^2|\log\varepsilon|^{-1}I_m+o(|\log\varepsilon|^{-1})$, where $b_α=(8-α)/2$.
