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Krea University（克雷亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究带多极Hardy势的分数阶Brezis-Nirenberg问题，证明在几何阈值与第一特征值之间临界Rayleigh商可达，从而存在正的最小能量解，并给出避免轮廓分解的紧致性估计。
AI 中文摘要
我们研究一个有界域上的分数阶Brezis-Nirenberg问题，其中包含有限多个以不同点$a_1,\dots,a_k$为中心的Hardy势，质量分别为$\mu_1,\dots,\mu_k\geq 0$。我们假设$\sum_{i=1}^k\mu_i<\Lambda_{N,s}$，并考虑$N>4s$和$0<\mu_{\max}<\overline{\mu}_{N,s}$。极点之间的相互作用允许在$\lambda=0$以及某些负值$\lambda$下解的存在性。在极点$a_j$处，这种相互作用由$B_j(\lambda,\mathbf a,\boldsymbol\mu)=\lambda+\sum_{i\neq j}\frac{\mu_i}{|a_i-a_j|^{2s}}$描述。这导致一个阈值$\lambda_{\mathrm{geo}}\leq 0$。我们证明当$\lambda_{\mathrm{geo}}<\lambda<\lambda_1(\boldsymbol\mu,\mathbf a)$时临界Rayleigh商可达，因此该问题存在一个正的最小能量解。如果至少有两个质量为正，则$\lambda_{\mathrm{geo}}<0$，因此结果包括$\lambda=0$和一个非空的负值$\lambda$区间。我们还通过局部化Gagliardo半范数，对弱消失序列证明了直接的紧致性估计。该论证避免了轮廓分解和Caffarelli-Silvestre延拓。
英文摘要
We study a fractional Brezis--Nirenberg problem on a bounded domain with finitely many Hardy potentials centered at distinct points $a_1,\dots,a_k$, with masses $μ_1,\dots,μ_k\geq 0$. We assume $\sum_{i=1}^kμ_i<Λ_{N,s}$ and consider $N>4s$ and $0<μ_{\max}<\overlineμ_{N,s}$. The interaction between the poles allows existence for $λ=0$ and for some negative values of $λ$. At a pole $a_j$, this interaction is described by $B_j(λ,\mathbf a,\boldsymbolμ)=λ+\sum_{i\neq j}\frac{μ_i}{|a_i-a_j|^{2s}}$. This leads to a threshold $λ_{\mathrm{geo}}\leq 0$. We prove that the critical Rayleigh quotient is attained for $λ_{\mathrm{geo}}<λ<λ_1(\boldsymbolμ,\mathbf a)$, and hence the problem admits a positive least-energy solution. If at least two masses are positive, then $λ_{\mathrm{geo}}<0$, so the result includes $λ=0$ and a nonempty interval of negative values of $λ$. We also prove a direct compactness estimate for weakly vanishing sequences by localizing the Gagliardo seminorm. The argument avoids both profile decomposition and the Caffarelli-Silvestre extension.
Comments24 pages
