Opuscula Math. 43, no. 1 (2023), 19-46

Opuscula Mathematica

Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions

Tomas Godoy

Abstract. Let \(\Omega\) be a \(C^{2}\) bounded domain in \(\mathbb{R}^{n}\) such that \(\partial\Omega=\Gamma_{1}\cup\Gamma_{2}\), where \(\Gamma_{1}\) and \(\Gamma_{2}\) are disjoint closed subsets of \(\partial\Omega\), and consider the problem\(-\Delta u=g(\cdot,u)\) in \(\Omega\), \(u=\tau\) on \(\Gamma_{1}\), \(\frac{\partial u}{\partial\nu}=\eta\) on \(\Gamma_{2}\), where \(0\leq\tau\in W^{\frac{1}{2},2}(\Gamma_{1})\), \(\eta\in(H_{0,\Gamma_{1}}^{1}(\Omega))^{\prime}\), and \(g:\Omega \times(0,\infty)\rightarrow\mathbb{R}\) is a nonnegative Carathéodory function. Under suitable assumptions on \(g\) and \(\eta\) we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow \(g\) to be singular at \(s=0\) and also at \(x\in S\) for some suitable subsets \(S\subset\overline{\Omega}\). The Dirichlet problem \(-\Delta u=g(\cdot,u)\) in \(\Omega\), \(u=\sigma\) on \(\partial\Omega\) is also studied in the case when \(0\leq\sigma\in W^{\frac{1}{2},2}(\Omega)\).

Keywords: singular elliptic problems, mixed boundary conditions, weak solutions.

Mathematics Subject Classification: 35J75, 35M12, 35D30.

Full text (pdf)

  1. R.A. Adams, Sobolev Spaces, Academic Press, New York, San Francisco, London, 1975.
  2. R.P. Agarwal, K. Perera, D. O'Regan, A variational approach to singular quasilinear problems with sign changing nonlinearities, Appl. Anal. 85 (2006), no. 10, 1201-1206. https://doi.org/10.1080/00036810500474655
  3. I. Bachar, H. Mâagli, V. Rădulescu, Singular solutions of a nonlinear elliptic equation in a punctured domain, Electron. J. Qual. Theory Differ. Equ. 2017, Paper no. 94. https://doi.org/10.14232/ejqtde.2017.1.94
  4. B. Bougherara, J. Giacomoni, Existence of mild solutions for a singular parabolic equation and stabilization, Adv. Nonlinear Anal. 4 (2015), no. 2, 123-134. https://doi.org/10.1515/anona-2015-0002
  5. B. Bougherara, J. Giacomoni, J. Hernández, Existence and regularity of weak solutions for singular elliptic problems, Proceedings of the 2014 Madrid Conference on Applied Mathematics in honor of Alfonso Casal, 19-30, Electron. J. Differ. Equ. Conf. 22, Texas State Univ., San Marcos, TX, 2015.
  6. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
  7. H. Brezis, X. Cabre, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 1 (1998), no. 2, 223-262.
  8. A. Callegari, A. Nachman, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math. 38 (1980), no. 2, 275-281. https://doi.org/10.1137/0138024
  9. Y. Chu, Y. Gao, W. Gao, Existence of solutions to a class of semilinear elliptic problem with nonlinear singular terms and variable exponent, J. Funct. Spaces (2016), Art. ID 9794739. https://doi.org/10.1155/2016/9794739
  10. F. Cîrstea, M. Ghergu, V. Rădulescu, Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type, J. Math. Pures Appl. 84 (2005), no. 4, 493-508. https://doi.org/10.1016/j.matpur.2004.09.005
  11. M.M. Coclite, G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations 14 (1989), no. 10, 1315-1327. https://doi-org.am.e-nformation.ro/10.1080/03605308908820656
  12. D.S. Cohen, H.B. Keller, Some positive problems suggested by nonlinear heat generators, J. Math. Mech. 16 (1967), 1361-1376.
  13. M.G. Crandall, P.H. Rabinowitz, L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), no. 2, 193-222. https://doi-org.am.e-nformation.ro/10.1080/03605307708820029
  14. M. Cuesta, P. Takáč, A strong comparison principle for positive solutions of degenerate elliptic equations, Differential Integral Equations 13 (2000), no. 4-6, 721-746.
  15. M.A. del Pino, A global estimate for the gradient in a singular elliptic boundary value problem, Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), no. 3-4, 341-352. https://doi.org/10.1017/S0308210500021144
  16. J.I. Diaz, J. Hernández, Positive and free boundary solutions to singular nonlinear elliptic problems with absorption: an overview and open problems, Electron. J. Differ. Equ. Conf. 21 (2014), 31-44.
  17. J.I. Diaz, J. Hernandez, J.M. Rakotoson, On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms, Milan J. Math. 79 (2011), Article no. 233. https://doi.org/10.1007/s00032-011-0151-x
  18. J.I. Diaz, J.M. Morel, L. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations 12 (1987), no. 12, 1333-1344. https://doi.org/10.1080/03605308708820531
  19. L. Dupaigne, M. Ghergu, V. Rădulescu, Lane-Emden-Fowler equations with convection and singular potential, J. Math. Pures Appl. 87 (2007), no. 6, 563-581. https://doi.org/10.1016/j.matpur.2007.03.002
  20. P. Feng, On the structure of positive solutions to an elliptic problem arising in thin film equations, J. Math. Anal. Appl. 370 (2010), no. 2, 573-583. https://doi.org/10.1016/j.jmaa.2010.04.049
  21. W. Fulks, J.S. Maybee, A singular nonlinear equation, Osaka J. Math. 12 (1960), 1-19.
  22. L. Gasiński, N.S. Papageorgiou, Nonlinear elliptic equations with singular terms and combined nonlinearities, Ann. Henri Poincaré 13 (2012), no. 3, 481-512. https://doi.org/10.1007/s00023-011-0129-9
  23. M. Ghergu, V.D. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and its Applications, vol. 37, The Clarendon Press, Oxford University Press, Oxford, 2008.
  24. M. Ghergu, V.D. Rădulescu, Nonlinear PDEs. Mathematical Models in Biology, Chemistry and Population Genetics, Springer-Verlag, Berlin, Heidelberg, New York, 2012.
  25. M. Ghergu, V.D. Rădulescu, Bifurcation and asymptotics for the Lane-Emden-Fowler equation, C. R. Math. Acad. Sci. Paris 337 (2003), no. 4, 259-264. https://doi.org/10.1016/S1631-073X(03)00335-2
  26. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, New York, 2001.
  27. T. Godoy, Strong solutions for singular Dirichlet elliptic problems, Electron. J. Qual. Theory Differ. Equ. 2022, Paper no. 40, 1-20. https://doi.org/10.14232/ejqtde.2022.1.40
  28. T. Godoy, A. Guerin, Positive weak solutions of elliptic Dirichlet problems with singularities in both the dependent and the independent variables, Electron. J. Qual. Theory Differ. Equ. 2019, Paper no. 54, 1-17. https://doi.org/10.14232/ejqtde.2019.1.73
  29. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Advanced Publishing Program, Marshfield, Massachusetts, 1985.
  30. J. Janus, J. Myjak, A generalizad Emden-Fowler equation with a negative exponent, Nonlinear Anal. 23 (1994), no. 8, 953-970. https://doi.org/10.1016/0362-546X(94)90193-7
  31. A.C. Lazer, P.J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc. 111 (1991), no. 3, 721-730. https://doi.org/10.2307/2048410
  32. Q. Li, W. Gao, Existence of weak solutions to a class of singular elliptic equations, Mediterr. J. Math. 13 (2016), no. 6, 4917-4927. https://doi.org/10.1007/s00009-016-0782-9
  33. H. Mâagli, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem, Nonlinear Anal. 74 (2011), no. 9, 2941-2947. https://doi.org/10.1016/j.na.2011.01.011
  34. L. Ma, J.C. Wei, Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal. 254 (2008), no. 4, 1058-1087. https://doi.org/10.1016/j.jfa.2007.09.017
  35. M. Montenegro, A.C. Ponce, The sub-supersolution method for weak solutions, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2429-2438.
  36. F. Oliva, F. Petitta, Finite and infinite energy solutions of singular elliptic problems: existence and uniqueness, J. Differential Equations 264 (2018), no. 1, 311-340. https://doi.org/10.1016/j.jde.2017.09.008
  37. N.S. Papageorgiou, G. Smyrlis, Nonlinear elliptic equations with singular reaction, Osaka J. Math. 53 (2016), no. 2, 489-514.
  38. N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, Nonlinear Analysis, Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019.
  39. K. Perera, E.A. Silva, Multiple positive solutions of singular \(p\)-Laplacian problems via variational methods, [in:] Differential & Difference Equations and Applications, Hindawi Publ. Corp., New York, 2006, pp. 915-924.
  40. K. Perera, E.A. Silva, On singular \(p\)-Laplacian problems, Differential Integral Equations 20 (2007), no. 1, 105-120.
  41. V.D. Rădulescu, Singular phenomena in nonlinear elliptic problems. From blow-up boundary solutions to equations with singular nonlinearities, [in:] Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 4, Elsevier/North-Holland, Amsterdam, 2007, pp. 483-591.
  42. J. Sabina de Lis, Hopf maximum principle revisited, Electron. J. Differential Equations 2015, no. 115, 9 pp.
  43. S. Salsa, Partial Differential Equations in Action - From Modelling to Theory, Springer, Milan, Berlin, Heidelberg, New York, 2008.
  44. J. Shi, M. Yao, On a singular nonlinear semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 6, 1389-1401. https://doi.org/10.1017/S0308210500027384
  45. J. Shi, M. Yao, Positive solutions for elliptic equations with singular nonlinearity, Electron. J. Qual. Theory Differ. Equ. 2005, no. 04, 11 pp.
  46. C.A. Stuart, Existence and approximation of solutions of nonlinear elliptic problems, Mathematics Report, Battelle Advanced Studies Center, Geneva, Switzerland, vol. 86, 1976.
  47. S. Yijing, Z. Duanzhi, The role of the power 3 for elliptic equations with negative exponents, Calc. Var. Partial Differential Equations 49 (2014), no. 3-4, 909-922. https://doi.org/10.1007/s00526-013-0604-x
  48. Z. Zhang, The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equation, J. Math. Anal. Appl. 312 (2005), no. 1, 33-43. https://doi.org/10.1016/j.jmaa.2005.03.023
  • Tomas Godoy
  • ORCID iD https://orcid.org/0000-0002-8804-9137
  • Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, Av. Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina
  • Communicated by Vicentiu D. Rădulescu.
  • Received: 2022-08-07.
  • Revised: 2022-11-21.
  • Accepted: 2022-11-25.
  • Published online: 2022-12-30.
Opuscula Mathematica - cover

Cite this article as:
Tomas Godoy, Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions, Opuscula Math. 43, no. 1 (2023), 19-46, https://doi.org/10.7494/OpMath.2023.43.1.19

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.