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

Abstract

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 \[ Δ^2u=\varepsilon^{8-α}K(x)e^{u(x)} \left(\int_Ω\frac{K(y)e^{u(y)}}{|x-y|^α}\,dy\right), \qquad u=Δu=0\quad\text{on }\partialΩ. \] Let $G$ be the Navier Green function, let $H$ be its regular part, and put $M_{\alpha }=8{\pi }^2(8-\alpha )$. 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 ${\epsilon }^{-1}$ and whose nonlinear source converges to $M_{\alpha }{\sum }_i{\delta }_{{\xi }_i^{\ast }}$. 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 \[ \operatorname{ind}(u_\varepsilon) =m+\operatorname{ind}\!\left(-D^2\mathcal F_m(\boldsymbolξ^*)\right). \] A critical four-dimensional capacity controls the scale directions. The dilation block of the reduced Hessian is positive and equals $8{\pi }^2{b}_{\alpha }^2|\log \epsilon {|}^{-1}{I}_m+o(|\log \epsilon {|}^{-1})$, where $b_{\alpha }=(8-\alpha )/2$.