Classification of solutions for the planar isotropic $L_p$ dual Minkowski problem

In his beautiful paper [1], Ben Andrews obtained the complete classification of the solutions of the planar isotropic $L_p$ Minkowski problem. In this paper, by generalizing Ben Andrews’s result we obtain the complete classification of the solutions of the planar isotropic $L_p$ dual Minkowski problem, that is, for any $p,q\in \mathbb{R}$ we obtain the complete classification of the solutions of the following equation: \begin{equation*} u^{1-p}(u_θ^2+u^2)^{\frac{q-2}{2}}(u_{θθ}+u)=1\quad$\text{on}$\ $\mathbb{S}$^1. \end{equation*} To establish the classification, we convert the ODE for the solution into an integral and study its asymptotic behavior, duality and monotonicity.