Multiple solutions of the planar $$L_p$$ dual Minkowski problem

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 $$\sqrt{q-p}-2$$ .