Function Space and Variational Problem for a Differential Equation with Degeneration

We introduce a function space on the positive real axis (0,∞) with multiweighted derivatives. A multiweighted derivative is defined as a successive differentiation procedure in which each derivative is taken after multiplying the function by a corresponding weight. The weight functions are assumed to be sufficiently smooth. We study the fundamental properties of this space and obtain weighted estimates for intermediate derivatives. Based on these results, we prove an existence and uniqueness theorem for generalized solutions of the Euler differential equation in this space for various types of degeneration on the boundary of the positive real axis. The solution is understood in the sense of the corresponding variational problem.