Abstract
Let u be a positive singular solution to boundary semilinear elliptic problems with a gradient term and a possibly singular nonlinearity. We prove the symmetry and monotonicity of u via the moving plane procedure.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Abdellaoui, B., Attar, A., Miri, S.E.: Nonlinear singular elliptic problem with gradient term and general datum. J. Math. Anal. Appl. 409(1), 362–377 (2014). https://doi.org/10.1016/j.jmaa.2013.07.017
Alexandrov, A.D.: A characteristic property of spheres. Ann. Mat. Pura Appl. 4(58), 303–315 (1962). https://doi.org/10.1007/BF02413056
Arcoya, D., Boccardo, L., Leonori, T., Porretta, A.: Some elliptic problems with singular natural growth lower order terms. J. Differ. Equ. 249(11), 2771–2795 (2010). https://doi.org/10.1016/j.jde.2010.05.009
Arcoya, D., Carmona, J., Leonori, T., Martínez-Aparicio, P.J., Orsina, L., Petitta, F.: Existence and nonexistence of solutions for singular quadratic quasilinear equations. J. Differ. Equ. 246(10), 4006–4042 (2009). https://doi.org/10.1016/j.jde.2009.01.016
Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bol. Soc. Brasil. Mat. Bull. 22(1), 1–37 (1991). https://doi.org/10.1007/BF01244896
Boccardo, L., Orsina, L.: Semilinear elliptic equations with singular nonlinearities. Calc. Var. Partial Differ. Equ. 37(3–4), 363–380 (2010). https://doi.org/10.1007/s00526-009-0266-x
Caffarelli, L., Li, Y.Y., Nirenberg, L.: Some remarks on singular solutions of nonlinear elliptic equations. II. Symmetry and monotonicity via moving planes. In: Advances in Geometric Analysis, Volume 21 of Adv. Lect. Math. (ALM), pp. 97–105. Int. Press, Somerville (2012)
Canino, A., Degiovanni, M.: A variational approach to a class of singular semilinear elliptic equations. J. Convex Anal. 11(1), 147–162 (2004)
Canino, A., Esposito, F., Sciunzi, B.: On the Höpf boundary lemma for singular semilinear elliptic equations. J. Differ. Equ. 266(9), 5488–5499 (2019). https://doi.org/10.1016/j.jde.2018.10.039
Canino, A., Grandinetti, M., Sciunzi, B.: Symmetry of solutions of some semilinear elliptic equations with singular nonlinearities. J. Differ. Equ. 255(12), 4437–4447 (2013). https://doi.org/10.1016/j.jde.2013.08.014
Canino, A., Montoro, L., Sciunzi, B.: The moving plane method for singular semilinear elliptic problems. Nonlinear Anal. 156, 61–69 (2017). https://doi.org/10.1016/j.na.2017.02.009
Canino, A., Sciunzi, B.: A uniqueness result for some singular semilinear elliptic equations. Commun. Contemp. Math. 18(6), 1550084 (2016). https://doi.org/10.1142/S0219199715500844. (9)
Crandall, M.G., Rabinowitz, P.H., Tartar, L.: On a Dirichlet problem with a singular nonlinearity. Commun. Partial Differ. Equ. 2(2), 193–222 (1977). https://doi.org/10.1080/03605307708820029
Esposito, F., Farina, A., Sciunzi, B.: Qualitative properties of singular solutions to semilinear elliptic problems. J. Differ. Equ. 265(5), 1962–1983 (2018). https://doi.org/10.1016/j.jde.2018.04.030
Esposito, F., Sciunzi, B.: The moving plane method for doubly singular elliptic equations involving a first-order term. Adv. Nonlinear Stud. 21(4), 905–916 (2021). https://doi.org/10.1515/ans-2021-2151
Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68(3), 209–243 (1979). (http://projecteuclid.org/euclid.cmp/1103905359)
Han, Z.-C.: Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 8(2), 159–174 (1991). https://doi.org/10.1016/S0294-1449(16)30270-0
Klimsiak, T.: Semilinear elliptic equations with Dirichlet operator and singular nonlinearities. J. Funct. Anal. 272(3), 929–975 (2017). https://doi.org/10.1016/j.jfa.2016.10.029
Lazer, A.C., McKenna, P.J.: On a singular nonlinear elliptic boundary-value problem. Proc. Am. Math. Soc. 111(3), 721–730 (1991). https://doi.org/10.2307/2048410
Li, C.: Local asymptotic symmetry of singular solutions to nonlinear elliptic equations. Invent. Math. 123(2), 221–231 (1996). https://doi.org/10.1007/s002220050023
Mazzeo, R., Pacard, F.: A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis. J. Differ. Geom. 44(2), 331–370 (1996). (http://projecteuclid.org/euclid.jdg/1214458975)
Oliva, F., Petitta, F.: On singular elliptic equations with measure sources. ESAIM Control Optim. Calc. Var. 22(1), 289–308 (2016). https://doi.org/10.1051/cocv/2015004
Oliva, F., Petitta, F.: Finite and infinite energy solutions of singular elliptic problems: existence and uniqueness. J. Differ. Equ. 264(1), 311–340 (2018). https://doi.org/10.1016/j.jde.2017.09.008
Pucci, P., Serrin, J.: The Maximum Principle. Progress in Nonlinear Differential Equations and their Applications, vol. 73. Birkhäuser Verlag, Basel (2007)
Sciunzi, B.: On the moving plane method for singular solutions to semilinear elliptic equations. J. Math. Pures Appl. 108(1), 111–123 (2017). https://doi.org/10.1016/j.matpur.2016.10.012
Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304–318 (1971). https://doi.org/10.1007/BF00250468
Stuart, C.A.: Existence and approximation of solutions of non-linear elliptic equations. Math. Z. 147(1), 53–63 (1976). https://doi.org/10.1007/BF01214274
Terracini, S.: On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv. Differ. Equ. 1(2), 241–264 (1996)
Véron, L.: Singular solutions of some nonlinear elliptic equations. Nonlinear Anal. 5(3), 225–242 (1981). https://doi.org/10.1016/0362-546X(81)90028-6
Véron, L.: Singularities of Solutions of Second Order Quasilinear Equations. Pitman Research Notes in Mathematics Series, vol. 353. Longman, Harlow (1996)
Funding
This research is funded by University of Economics and Law, Vietnam National University, Ho Chi Minh City, Vietnam.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Le, P. Doubly Singular Elliptic Equations Involving a Gradient Term: Symmetry and Monotonicity. Results Math 79, 3 (2024). https://doi.org/10.1007/s00025-023-02025-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-023-02025-y
Keywords
- Semilinear elliptic equation
- singular solution
- singular nonlinearity
- gradient term
- symmetry and monotonicity