A Comparative Study of Two Weighted Function-Space Formulations for a Degenerate–Singular Elliptic Operator

The elliptic operator studied here combines a degenerate leading coefficient in one space variable with a singular lower-order coefficient in the other, and admits more than one natural variational treatment. We examine two of them side by side. The first places the degeneracy weight on the derivative in the degenerate direction alone, producing an anisotropically weighted Sobolev space; there we prove that the associated nonlinear Dirichlet problem admits a unique weak solution once the degeneracy exponent and the singular coefficient, together with the growth constant of a Lipschitz nonlinearity, stay below explicit thresholds; interior second-order regularity holds throughout the admissible range of the degeneracy exponent, and a conditional global weighted regularity statement is obtained in the complementary, stronger-degeneracy regime. The second treatment introduces an auxiliary weight, chosen independently of the degeneracy exponent and applied to both derivatives on top of the existing anisotropy; carrying the derivative of this auxiliary weight through the weak formulation produces a bilinear form whose singular term is governed by the shifted parameter $\alpha-\beta_0$ rather than by $\beta_0$ alone, and yields a coercivity condition requiring the absolute value of $\alpha-\beta_0$ to stay strictly below one half of $1-\alpha$ -- that is, an admissible interval for $\beta_0$ centered at $\alpha$ rather than at $0$, whose half-width shrinks linearly from one half down to zero as the auxiliary exponent $\alpha$ ranges from zero to one -- valid for every degeneracy exponent, shown by an explicit near-extremal sequence to be sharp on one side, and leading to well-posedness, stability estimates, and interior regularity away from the degenerate boundary. We then compare the two resulting thresholds quantitatively -- showing, in particular, that they coincide in an appropriate limiting regime -- and list the specific questions each formulation leaves open.