Degenerate equations of monotone type: Lavrent'ev phenomenon and attainability problems

A non-linear monotone equation with degenerate weight function is considered. In the general case the smooth functions are not dense in the corresponding weighted Sobolev space $W$, which leads to a non-uniqueness of a particular kind. Taking for the energy space either $W$ itself or its subspace $H$ equal to the closure of the smooth functions one obtains at least two uniquely soluble problems. In addition, there exist infinitely many weak solutions distinct from the $W$- and $H$-solutions. The problem of approximability or attainability is considered: which solutions of the original equation can be obtained as limits of solutions of the equations with suitable non-degenerate weights? It is shown that the $W$- and the $H$-solutions are attainable; in both cases a regular approximation algorithm is described.
Bibliography: 14 titles.