The Lₚ dual Minkowski problem for unbounded closed convex sets

The central focus of this paper is the Lₚ dual Minkowski problem for C-compatible sets, where C is a pointed closed convex cone in Rⁿ with nonempty interior. Such a problem deals with the characterization of the $(p, q)$-th dual curvature measure of a C-compatible set. It produces new Monge-Amp\`{e}re equations for unbounded convex hypersurface, often defined over open domains and with non-positive unknown convex functions. Within the family of C-determined sets, the Lₚ dual Minkowski problem is solved for 0≠ p∈ R and q∈ R; while it is solved for the range of p≤ 0 and $p<q$ within the newly defined family of $(C, p, q)$-close sets. When p≤ q, we also obtain some results regarding the uniqueness of solutions to the Lₚ dual Minkowski problem for C-compatible sets.