On the solvability of a boundary value problem for a quasilinear equation of mixed type with two degeneration lines

The article investigates the existence of a generalized solution to one boundary value problem for an equation of mixed type with two lines of degeneration in the weighted space of S.L. Sobolev. In proving the existence of a generalized solution, the spaces of functions U(Ω) and V (Ω) are introduced, the spaces H 1 (Ω) and H 1 * (Ω) are defined as the completion of these spaces of functions, respectively, with respect to the weighted norms, including the functions K( y ) and N( x ). Using an auxiliary boundary value problem for a first order partial differential equation, Kondrashov’s theorem on the compactness of the embedding of W 2 1 (Ω) in L 2 (Ω) and Vishik’s lemma, the existence of a solution to the boundary value problem is proved.