On the classification of solutions to the Logarithmic Laplacian critical Choquard equation

Abstract

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 \[ B_{N,0} =\left(\frac{Γ(N)}{π^{N/2}Γ(N/2)}\right)^{1/2}, \qquad B_{N,\mathcal{L}} =2\ln 2+4ψ(N/2)-2ψ(N)+\frac{4}{N}\ln B_{N,0}. \]