We are concerned with the planar \(L_p\) dual Minkowski problem with indices p, q. Through the compactness analysis of an associated constrained variational problem in Sobolev space, the solvability of the planar \(L_p\) dual Minkowski problem and the related functional inequality are established, upon which the multiple solutions to the planar \(L_p\) dual Minkowski problem are obtained. Precisely, if \(q\ge 2\) is even, \(p<0\) and \(q-p>16\), there exist at least \([\sqrt{q-p}-2\ ]\) convex bodies whose \(L_p\) dual curvature measure is equal to the standard spherical measure in the plane, where \([\sqrt{q-p}-2\ ]\) is the integer part of \(\sqrt{q-p}-2\).

我们关注指数为p，  q的平面\（L_p \）对偶Minkowski问题。通过对Sobolev空间中一个相关约束变分问题的紧致性分析，建立了平面\（L_p \）对偶Minkowski问题的可解性和相关的函数不等式，从而对平面\（L_p \）对偶Minkowski的多重解获得问题。精确地讲，如果\（q \ ge 2 \）是偶数，\（p <0 \）和\（qp> 16 \），则至少存在\（[[\ sqrt {qp} -2 \] \）个凸体谁的\（L_p \）对偶曲率度量等于平面中的标准球面度量，其中\（[\ sqrt {qp} -2 \] \）是\（\ sqrt {qp} -2 \）的整数部分。