In this paper, we study the planar $L_p$-Minkowski problem(0.1)$u_{\theta \theta }+u=f{u}^{p-1},\theta \in {\mathbb{S}}^1$ for all $p\in \mathbb{R}$, which was introduced by Lutwak [21]. A detailed exploration of (0.1) on solvability will be presented. More precisely, we will prove that for $p\in (0,2)$, there exists a positive function $f\in C^{\alpha }({\mathbb{S}}^1),\alpha \in (0,1)$ such that (0.1) admits a nonnegative solution vanishes somewhere on ${\mathbb{S}}^1$. In case $p\in (-1,0]$, a surprising a-priori upper/lower bound for solution was established, which implies the existence of positive classical solution to each positive function $f\in C^{\alpha }({\mathbb{S}}^1)$. When $p\in (-2,-1]$, the existence of some special positive classical solution has already been known using the Blaschke-Santalo inequality [7]. Upon the final case $p\le -2$, we show that there exist some positive functions $f\in C^{\alpha }({\mathbb{S}}^1)$ such that (0.1) admits no solution. Our results clarify and improve largely the planar version of Chou-Wang's existence theorem [7] for $p<2$. At the end of this paper, some new uniqueness results will also be shown.

在本文中，我们研究平面 ${\mathrm{大号}}_p$-Minkowski问题（0.1）${ü}_{\theta \theta }+ü=F{ü}^{p-\mathrm{1个}}，\theta \in {\mathbb{小号}}^{\mathrm{1个}}$ 对全部 $p\in \mathbb{[R}$，这是由Lutwak [21]提出的。将介绍（0.1）关于可溶性的详细说明。更确切地说，我们将证明$p\in （0，\mathrm{2个}）$，有一个积极的作用 $F\in C^{\alpha }（{\mathbb{小号}}^{\mathrm{1个}}），\alpha \in （0，\mathrm{1个}）$ 使得（0.1）承认非负解消失在 ${\mathbb{小号}}^{\mathrm{1个}}$。以防万一$p\in （-\mathrm{1个}，0]$，建立了令人惊讶的先验上/下界解，这意味着每个正函数都存在正经典解$F\in C^{\alpha }（{\mathbb{小号}}^{\mathrm{1个}}）$。什么时候$p\in （-\mathrm{2个}，-\mathrm{1个}]$，已经使用Blaschke-Santalo不等式[7]知道了一些特殊的正经典解的存在。在最后的情况下$p\le -\mathrm{2个}$，我们发现这里存在一些积极的功能 $F\in C^{\alpha }（{\mathbb{小号}}^{\mathrm{1个}}）$因此（0.1）不接受任何解决方案。我们的结果在很大程度上澄清和改进了Chou-Wang存在定理[7]的平面形式。$p<\mathrm{2个}$。在本文的结尾，还将显示一些新的唯一性结果。