Concentration, Local Uniqueness, and Morse Index of Multi-bubble Solutions for a Critical Exponential Biharmonic Choquard Equation

Let $\Omega\subset\mathbb R^4$ be a bounded domain of class $C^6$, let $0<\alpha<4$, and let $K\in C^4(\overline\Omega)$ be positive. We study the Navier problem $$\Delta^2u=\varepsilon^{8-\alpha}K(x)e^{u(x)} \left(\int_\Omega\frac{K(y)e^{u(y)}}{|x-y|^\alpha}\,dy\right), \qquad u=\Delta u=0\quad\text{on }\partial\Omega.$$ Let $G$ be the Navier Green function, let $H$ be its regular part, and put $M_\alpha=8\pi^2(8-\alpha)$. The concentration points are governed by $$\mathcal F_m(\boldsymbol\xi) =\sum_{i=1}^m \left[\log K(\xi_i)+\frac{M_\alpha}{2}H(\xi_i,\xi_i)\right] +M_\alpha\sum_{i<j}G(\xi_i,\xi_j).$$ 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 $M_\alpha\sum_i\delta_{\xi_i^*}$. 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 $H^2(\Omega)\cap H_0^1(\Omega)$, and $$\operatorname{ind}(u_\varepsilon) =m+\operatorname{ind}\!\left(-D^2\mathcal F_m(\boldsymbol\xi^*)\right).$$ A critical four-dimensional capacity controls the scale directions. The dilation block of the reduced Hessian is positive and equals $8\pi^2b_\alpha^2|\log\varepsilon|^{-1}I_m+o(|\log\varepsilon|^{-1})$, where $b_\alpha=(8-\alpha)/2$.