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Doubly Singular Elliptic Equations Involving a Gradient Term: Symmetry and Monotonicity

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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.

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References

  1. 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

    Article  MathSciNet  Google Scholar 

  2. Alexandrov, A.D.: A characteristic property of spheres. Ann. Mat. Pura Appl. 4(58), 303–315 (1962). https://doi.org/10.1007/BF02413056

    Article  MathSciNet  Google Scholar 

  3. 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

    Article  MathSciNet  Google Scholar 

  4. 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

    Article  MathSciNet  Google Scholar 

  5. 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

    Article  MathSciNet  Google Scholar 

  6. 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

    Article  MathSciNet  Google Scholar 

  7. 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)

    Google Scholar 

  8. Canino, A., Degiovanni, M.: A variational approach to a class of singular semilinear elliptic equations. J. Convex Anal. 11(1), 147–162 (2004)

    MathSciNet  Google Scholar 

  9. 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

    Article  Google Scholar 

  10. 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

    Article  MathSciNet  Google Scholar 

  11. 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

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

  13. 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

    Article  MathSciNet  Google Scholar 

  14. 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

    Article  MathSciNet  Google Scholar 

  15. 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

    Article  MathSciNet  Google Scholar 

  16. 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)

    Article  MathSciNet  Google Scholar 

  17. 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

    Article  MathSciNet  Google Scholar 

  18. 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

    Article  MathSciNet  Google Scholar 

  19. 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

    Article  MathSciNet  Google Scholar 

  20. 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

    Article  MathSciNet  Google Scholar 

  21. 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)

    Article  MathSciNet  Google Scholar 

  22. 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

    Article  MathSciNet  Google Scholar 

  23. 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

    Article  MathSciNet  Google Scholar 

  24. Pucci, P., Serrin, J.: The Maximum Principle. Progress in Nonlinear Differential Equations and their Applications, vol. 73. Birkhäuser Verlag, Basel (2007)

    Google Scholar 

  25. 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

    Article  MathSciNet  Google Scholar 

  26. Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304–318 (1971). https://doi.org/10.1007/BF00250468

    Article  MathSciNet  Google Scholar 

  27. 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

    Article  MathSciNet  Google Scholar 

  28. 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)

    MathSciNet  Google Scholar 

  29. 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

    Article  MathSciNet  Google Scholar 

  30. Véron, L.: Singularities of Solutions of Second Order Quasilinear Equations. Pitman Research Notes in Mathematics Series, vol. 353. Longman, Harlow (1996)

    Google Scholar 

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Funding

This research is funded by University of Economics and Law, Vietnam National University, Ho Chi Minh City, Vietnam.

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  1. Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam

    Phuong Le

  2. Vietnam National University, Ho Chi Minh City, Vietnam

    Phuong Le

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  1. Phuong Le
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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

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