发表机构
Indian Institute of Technology (IIT-BHU); LMAP, UMR E2S-UPPA CNRS 5142（印度理工学院（贝拿勒斯印度教大学）; 法国国家科学研究中心联合研究实验室LMAP）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立sharp对数Choquard不等式，并分类了临界对数Choquard方程的正经典解，证明其具有由气泡极限给出的显式形式。
AI 中文摘要
本文通过结合Beckner熵不等式与sharp Pitt型不等式，建立了sharp对数Choquard不等式。受此估计的启发，我们在适当的可积性类中，对临界对数Choquard问题\\[ \mathcal{L}_\Delta u = \sigma u + \frac{1}{\\|u\\|_2^2} \left(G_{\ln}(u)-\frac{4}{N}\int_{\mathbb{R}^N}u^2\ln u\\,dx\right)u \qquad\text{in }\mathbb{R}^N, \\] 的正经典解进行分类，其中$\sigma\in\mathbb{R}$，$N\geq 1$，$\mathcal{L}_\Delta$为对数拉普拉斯算子，$G_{\ln}(u)=\ln(1/|x|^4)*u^2$。我们构造了作为临界分数Choquard气泡极限的显式解，并证明了所给类中的每个正经典解都具有形式\\[ u_{\sigma,t}(x) = e^{\frac{N}{4}(\sigma-B_{N,\mathcal{L}})} B_{N,0}\left(\frac{t}{t^2+|x-x_0|^2}\right)^{N/2}, \qquad t>0,\quad x_0\in\mathbb{R}^N, \\] 其中\\[ B_{N,0} =\left(\frac{\Gamma(N)}{\pi^{N/2}\Gamma(N/2)}\right)^{1/2}, \qquad B_{N,\mathcal{L}} =2\ln 2+4\psi(N/2)-2\psi(N)+\frac{4}{N}\ln B_{N,0}. \\]
英文摘要
In this work, we establish a sharp logarithmic Choquard inequality by combining Beckner's entropy inequality with the sharp Pitt-type inequality. Motivated by this estimate, we classify the positive classical solutions, in a suitable integrability class, of the critical logarithmic Choquard problem \[ \mathcal{L}_Δu = σu + \frac{1}{\|u\|_2^2} \left(G_{\ln}(u)-\frac{4}{N}\int_{\mathbb{R}^N}u^2\ln u\,dx\right)u \qquad\text{in }\mathbb{R}^N, \] where $σ\in\mathbb{R}$, $N\geq 1$, $\mathcal{L}_Δ$ is the logarithmic Laplacian, and $G_{\ln}(u)=\ln(1/|x|^4)*u^2$. We construct explicit solutions as limits of critical fractional Choquard bubbles and prove that every positive classical solution in the prescribed class has the form \[ u_{σ,t}(x) = e^{\frac{N}{4}(σ-B_{N,\mathcal{L}})} B_{N,0}\left(\frac{t}{t^2+|x-x_0|^2}\right)^{N/2}, \qquad t>0,\quad x_0\in\mathbb{R}^N, \] where \[ B_{N,0} =\left(\frac{Γ(N)}{π^{N/2}Γ(N/2)}\right)^{1/2}, \qquad B_{N,\mathcal{L}} =2\ln 2+4ψ(N/2)-2ψ(N)+\frac{4}{N}\ln B_{N,0}. \]
Comments24 pages. Comments are welcome
