The two dimensional \(L_p\) Minkowski problem and nonlinear equations with negative exponents
The \(L_p\) Minkowski problem asks when a given Borel measure on the sphere \(S^{n-1}\) can be the \(L_p\) surface area measure of a convex body \(K\). In the plane, this question is equivalent to the following analytical one: given a function \(g(\theta)\), are there positive solutions of the differential equation NEWLINE\[NEWLINE u''(\theta)+u(\theta)=g(\theta)u(\theta)^{p-1}\;\text{ on }\;S^1? NEWLINE\]NEWLINE In [Adv. Math. 201, No. 1, 77--89 (2006; Zbl 1102.34023)], \textit{W. Chen} obtained certain existence results for this problem when \(-2\leq p\leq 0\). In the paper under review the authors extend these results for all \(p\leq -2\). They also consider the related conformal curvature problem with opposite curvature sign, i.e., they give conditions for the existence of positive solutions of the differential equation \(-u''(\theta)+u(\theta)=g(\theta)u(\theta)^{p-1}\), \(p\leq 0\), on the 1-dimensional sphere.
scientific article; zbMATH DE number 6059027
scientific article