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

In a previous paper [H. Li and B. Xu, arXiv:2211.06875 (2022)], 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 φ−pφθθ−φθ2/(2φ)+(φ−φ−1)/2=γ on S1, where γ is a positive constant. We provide a classification of solutions 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.