Function space and variational problem for a differential equation with degeneracy

In this paper, a functional space with multiweighted derivatives is introduced on the half-line $(0, \infty)$. A multiweighted derivative is understood as the operation of successive differentiation, where each derivative is taken of a function previously multiplied by the corresponding weight function. It is assumed that the weight functions possess sufficient smoothness. The main properties of the introduced space are investigated, and weighted estimates for intermediate derivatives are obtained. Based on these results, a theorem on the existence and uniqueness of a generalized solution to the Euler-type differential equation belonging to the introduced space is proved for various types of degeneracy at the boundary of the half-line. The solution is considered as a solution to the corresponding variational problem.