Existence and uniqueness of solutions to Liouville equation
Alireza Ataei
TL;DR
The paper advances the theory of the Liouville equation $-\,\Delta \psi = 4\pi \beta V e^{\psi}$ on $\mathbb{R}^2$ with $\int V e^{\psi}=1$ by establishing constructive existence results under minimal local integrability of $V$, providing sharp decay conditions and a variational framework that avoids many classical techniques. It proves a generalized comparison principle to obtain uniqueness for $\beta<0$ and a radial uniqueness result for $\beta>0$ under natural monotonicity assumptions on $V$, while highlighting non-uniqueness in the conformally invariant case. The work also develops a detailed blow-up analysis and demonstrates broad applicability to problems in differential geometry (Berger–Nirenberg), statistical mechanics (mean-field vortex models), and quantum mechanics (Chern–Simons-Schrödinger models), including the nonlinear Landau level regime and asymptotic behaviors as external fields vary. Overall, it provides a rigorous, constructive, and versatile treatment of existence, uniqueness, asymptotics, and applications of Liouville-type mean-field equations in two dimensions, with sharp conditions and new connections across geometry, analysis, and physics.
Abstract
We prove some general results on the existence and uniqueness of solutions to the Liouville equation. Then, we discuss the sharpness and possible generalizations. Finally, we give several applications, arising in both mathematics and physics.