A Comparative Study of Two Weighted Function-Space Formulations for a Degenerate--Singular Elliptic Operator
DOI:
https://doi.org/10.56143/ujmcs.v2i2.5Keywords:
degenerate elliptic operator; Bessel-type singular coefficient; weighted Sobolev space; Hardy-type inequality; coercivity threshold; Galerkin approximation; Lipschitz nonlinearity.Abstract
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.
References
[1] Chaplain, M.A.J., Preziosi, L.: Mathematical Oncology. Springer, Cham (2025).
[2] Keldysh, M.V.: On certain cases of degeneration of equations of elliptic type on the boundary of a domain. Dokl. Akad. Nauk SSSR 77(2), 181–183 (1951) (in Russian).
[3] Kipriyanov, I.A.: Singular Elliptic Boundary Value Problems. Nauka, Moscow (1997) (in Russian).
[4] Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations 32(8), 1245–1260 (2007). https://doi.org/10.1080/03605300600987306
[5] Salakhitdinov, M.S., Mirsaburov, M.: Nelokal’nye zadachi dlia uravnenii smeshannogo tipa s singuliarnymi koeffitsientami. Tashkent University Press, Tashkent (2005).
[6] Mirsaburov, M.: A boundary value problem for a class of mixed-type equations with the Bitsadze–Samarskii condition on degeneration lines. Uzbek Math. J. 2, 48–57 (2012).
[7] Mirsaburov, M., Ruziev, M.Kh.: A problem with conditions on both characteristics and on a degeneration line for a class of hyperbolic–parabolic equations. Differ. Equ. 50, 1347–1359 (2014).
[8] Aharrouch, B., Bennouna, J.: Nonlinear degenerate elliptic equations in weighted Sobolev spaces. Electron. J. Differential Equations 2020, No. 105 (2020).
[9] El Ouaarabi, M., Allalou, C., Melliani, S., et al.: On a class of nonlinear degenerate elliptic equations in weighted Sobolev spaces. Georgian Math. J. 30(1), 81–94 (2023). https://doi.org/10.1515/gmj-2022-2191
[10] Alimohammady, M., Cattani, C., Koozehgar Kalleji, M.: Existence and regularity results for a degenerate elliptic nonlinear equation on weighted Sobolev spaces. Quaestiones Mathematicae 47(1), 157–171 (2024). https://doi.org/10.2989/16073606.2023.2205033
[11] Cavalheiro, A.C.: Weighted Sobolev Spaces and Degenerate Elliptic Equations. Cambridge Scholars Publishing, Newcastle upon Tyne (2023).
[12] Beisebay, P., Akzhigitov, Y., Akhazhanov, T., Kenzhebekova, G., Matin, D.: Smoothness of the solution of a boundary value problem for degenerate elliptic equations. Symmetry 17(9), 1562 (2025). https://doi.org/10.3390/sym17091562
[13] Di Fazio, G., Fanciullo, M.S., Zamboni, P.: Regularity up to the boundary for some degenerate elliptic operators. Applicable Analysis 101(10), 3563–3575 (2022). https://doi.org/10.1080/00036811.2020.1769079
[14] Do, T.D.: Harnack inequality for degenerate Hormander sub-elliptic operators with potentials. Expositiones Mathematicae 40(3), 605–627 (2022). https://doi.org/10.1016/j.exmath.2021.12.004
[15] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165, 207–226 (1972).
[16] Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations 7(1), 77–116 (1982). https://doi.org/10.1080/03605308208820218
[17] Balci, A.Kh., Diening, L., Giova, R., Passarelli di Napoli, A.: Elliptic equations with degenerate weights. SIAM J. Math. Anal. 54(2), 2373–2412 (2022). https://doi.org/10.1137/21M1412529
[18] Opic, B., Kufner, A.: Hardy-Type Inequalities. Pitman Research Notes in Mathematics Series, vol. 219. Longman, Harlow (1990).
[19] Wang, B.-S., Hou, G.-L., Ge, B.: Existence and uniqueness of solutions for the p(x)-Laplacian equation with convection term. Mathematics 8(10), 1768 (2020). https://doi.org/10.3390/math8101768
[20] Tiarimti Alaoui, A., Jourhmane, M.: Existence and uniqueness of weak solutions for novel anisotropic nonlinear diffusion equations related to image analysis. J. Math. 2021, Art. 5553126 (2021). https://doi.org/10.1155/2021/5553126
[21] Ghaffari Laleh, N., Loeffler, C.M.L., Grajek, J., et al.: Classical mathematical models for prediction of response to chemotherapy and immunotherapy. PLoS Comput. Biol. 18(2), e1009822 (2022). https://doi.org/10.1371/journal.pcbi.1009822
[22] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin–Heidelberg (2001).
[23] Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985).
