Existence of solutions to the generalized dual Minkowski problem

Given a real number q and a star body in the n-dimensional Euclidean space, the generalized dual curvature measure of a convex body was introduced by Lutwak-Yang-Zhang 43. The corresponding generalized dual Minkowski problem is studied in this paper. By using variational methods, we solve the generalized dual Minkowski problem for q<0, and the even generalized dual Minkowski problem for 0 q1. We also obtain a sufficient condition for the existence of solutions to the even generalized dual Minkowski problem for 1<q<n.