Existence of solutions to the even Gaussian dual Minkowski problem

In this paper, we consider the Gaussian dual Minkowski problem. The problem involves a new type of fully nonlinear partial differential equations on the unit sphere. Our main purpose is to show the existence of solutions to the even Gaussian dual Minkowski problem for $q>0$. More precisely, we will show that there exists an origin-symmetric convex body K in $R^n$ such that its Gaussian dual curvature measure ${\tilde{C}}_{{\gamma }_n,q}(K,\cdot )$ has density f (up to a constant) on the unit sphere when $q>0$ and f has positive upper and lower bounds. Note that if f is smooth then K is also smooth. As the application of smooth solutions, we completely solve the even Gaussian dual Minkowski problem for $q>0$ based on an approximation argument.