Classification of solutions to the isotropic horospherical p-Minkowski problem in hyperbolic plane

In {LX}, the first author and Xu introduced and studied the horospherical p-Minkowski problem in hyperbolic space Hⁿ⁺¹. In particular, they established the uniqueness result for solutions to this problem when the prescribed function is constant and p≥ -n. This paper focuses on the isotropic horospherical p-Minkowski problem in hyperbolic plane H², which corresponds to the equation {equation}{0} φ⁻ᵖ(φθθ-{φθ^2}{2φ}+{φ-φ⁻¹}{2})=γ\ S^1, {equation} where γ is a positive constant. We provide a classification of solutions to the above equation for p≥ -7, as well as a nonuniqueness result of solutions for $p<-7$. Furthermore, we extend this problem to the isotropic horospherical q-weighted p-Minkowski problem in hyperbolic plane and derive some uniqueness and nonuniqueness results.