AI 中文总结
研究\(\mathbb{R}^3\)中含奇异哈代势的临界薛定谔 - 泊松系统，通过结合哈代不等式、约束变分法等，解决逆平方哈代奇点等带来的困难，得到归一化基态及多个解，并证明半经典状态在\(\varepsilon\to0\)时集中在电势全局最小集附近。
AI 中文摘要
本文研究了\(\mathbb{R}^3\)中具有奇异哈代势的临界薛定谔 - 泊松系统归一化解的存在性、多重性和半经典集中情况。具体考虑了在规定质量约束\(\int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3\)下的方程组。主要困难源于逆平方哈代奇点、质量约束、临界局部非线性和非局部泊松相互作用同时存在。通过多种方法，首先为足够小的质量和\(\varepsilon\)建立了归一化基态的存在性，然后利用理论获得多个归一化解，其数量与\(V\)的最小集拓扑有关，最后表明当\(\varepsilon\to0\)时，相应的半经典状态集中在电势的全局最小集附近。
英文摘要
In this paper, we investigate the existence, multiplicity, and semiclassical concentration of normalized solutions to a critical Schrödinger--Poisson system with a singular Hardy potential in \(\mathbb{R}^3\). More precisely, we consider \[ \begin{cases} -\varepsilon^2Δu+ \left(V(x)-\dfrac{κ\varepsilon^2}{|x|^2}\right)u -ϕ|u|^3u =λu+μ|u|^{q-2}u+|u|^4u, & \text{in } \mathbb{R}^3, \\[1mm] -\varepsilon^2Δϕ=|u|^5, & \text{in } \mathbb{R}^3, \end{cases} \] under the prescribed mass constraint \[ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \] where \(a,μ>0\), \(q\in(2,10/3)\), \(\varepsilon>0\) is a small semiclassical parameter, and \(0<κ<1/4\). The parameter \(λ\in\mathbb{R}\) appears as a Lagrange multiplier associated with the mass constraint, while \(V:\mathbb{R}^3\to(0,+\infty)\) is a continuous electric potential whose minimum set is assumed to be nonempty and compact. The main difficulty stems from the simultaneous presence of the inverse-square Hardy singularity, the mass constraint, the critical local nonlinearity, and the nonlocal Poisson interaction. By combining the Hardy inequality, constrained variational methods, suitable truncation arguments, and concentration-compactness techniques, we first establish the existence of a normalized ground state for sufficiently small mass and sufficiently small \(\varepsilon\). We then employ Ljusternik--Schnirelmann category theory to obtain multiple normalized solutions whose number is related to the topology of the minimum set of \(V\). Finally, we show that the corresponding semiclassical states concentrate near the global minimum set of the electric potential as \(\varepsilon\to0\).
