  This study aims to investigate the functional properties of weak solution
spaces and their compact embedding properties in relation to the Dirichlet
problem associated with a specific class of degenerate elliptic equations. To
expand the scope of analysis for degenerate elliptic problems, we employ
weighted Sobolev inequalities and compact embedding techniques within the
framework of weighted Sobolev spaces. Additionally, we apply these findings to
examine a Stackelberg-Nash game problem and obtain the existence of the
Stackelberg-Nash equilibrium.
