Multiple solutions of the planar \(L_p\) dual Minkowski problem
The authors study the planar \(L_p\) dual Minkowski problem, associated to the \(L_p\) dual curvature measures \(\tilde C_{p,q}(K,\cdot)\), of a convex body \(K\), on the unit sphere, for the case \(p<0\), and \(q\geq 2\) even. In the paper under review, the associated differential equation of the studied Minkowski problem is given by \[u''+u=g(\theta)u^{p-1}(u^2+|u'|^2)^{\frac{2-q}{2}},\quad \theta\in S^1,\] where \(g\) is the density function of a finite Borel measure \(\mu\) on \(S^1\). The authors first establish the existence of a weak solution of the above differential equation when \(g\) is a \(2\pi/m\)-periodic function with \(\int_{S^1}g(\theta)d\theta>0\) and \(m\geq 4\). The existence of multiple solutions is given for \(g(\theta)\equiv 1\), \(q-p>16\). In this case, it is proved that there are at least \(\lfloor\sqrt{q-p}-2\rfloor\) convex bodies with \(L_p\) dual curvature measure equal to the standard spherical measure. Moreover, the authors show that for an integer \(m\in [4,\sqrt{q-p})\), there exists a positive classical solution with the least period \(2\pi/m\). The authors use a variational method to prove the existence theorem and the compactness analysis of a variational problem to obtain a functional inequality associated to the above differential equation to prove the existence of multiple solutions.
scientific article; zbMATH DE number 7344737
scientific article