• Sun, Juntao (School of Science, Shandong University of Technology) ;
  • Chu, Jifeng (Department of Mathematics, College of Science, Hohai University)
  • Received : 2011.06.09
  • Published : 2013.07.01


In this paper, we establish the existence results for second order singular Dirichlet problems via variational methods. Some recent results are extended and improved. Examples are also given to illustrate the new results.


positive solutions;singular Dirichlet problems;variational methods;critical points


  1. R. P. Agarwal and D. O'Regan, Singular Differential and Integral Equations with Ap-plications, Kluwer Academic Publishers, Dordrecht, 2003.
  2. R. P. Agarwal and D. O'Regan, Singular boundary value problems for superlinear second order ordinary and delay differential equations, J. Differential Equations 130 (1996), no. 2, 333-355.
  3. R. P. Agarwal and D. O'Regan, Existence theory for single and multiple solutions to singular positone boundary value problems, J. Differential Equations 175 (2001), no. 2, 393-414.
  4. R. P. Agarwal and D. O'Regan, Existence criteria for singular boundary value problems with sign changing nonlinearities, J. Differential Equations 183 (2002), no. 2, 409-433.
  5. R. P. Agarwal, K. Perera, and D. O'Regan, Multiple positive solutions of singular problems by variational methods, Proc. Amer. Math. Soc. 134 (2006), no. 3, 817-824.
  6. R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Clarendon Press, 8, Oxford, 1975.
  7. J. V. Baxley, A singular nonlinear boundary value problem: membrane response of a spherical cap, SIAM J. Appl. Math. 48 (1988), no. 3, 497-505.
  8. G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl. 2009 (2009), Article ID 670675, 20 pages.
  9. G. Bonanno and B. Di Bella, Infinitely many solutions for a fourth-order elastic beam equation, Nonlinear Differential Equations Appl. 18 (2011), no. 3, 357-368.
  10. A. Callegari and A. Nachman, Some singular nonlinear differential equations arising in boundary layer theory, J. Math. Anal. Appl. 64 (1978), no. 1, 96-105.
  11. J. Chu and D. O'Regan, Multiplicity results for second order non-autonomous singular Dirichlet systems, Acta Appl. Math. 105 (2009), no. 3, 323-338.
  12. J. A . Cid, O. L. Pouso, and R. L. Pouso, Existence of infinitely many solutions for second-order singular initial value problems with an application to nonlinear massive gravity, Nonlinear Anal. Real World Appl. 12 (2011), no. 5, 2596-2606.
  13. L. Erbe and R. Mathsen, Positive solutions for singular nonlinear boundary value problems, Nonlinear Anal. 46 (2001), no. 7, 979-986.
  14. P. Habets and F. Zanolin, Upper and lower solutions for a generalized Emden-Fowler equation, J. Math. Anal. Appl. 181 (1994), no. 3, 684-700.
  15. X. He and W. Zou, Infinitely many solutions for a singular elliptic equation involving critical sobolev-Hardy exponents in $\mathbb{R}^n$, Acta Math. Sci. Ser. B Engl. Ed. 30 (2010), no. 3, 830-840.
  16. K. Lan, Multiple positive solutions of semilinear differential equations with singularities, J. London Math. Soc. (2) 63 (2001), no. 3, 690-704.
  17. K. Lan and J. R. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations 148 (1998), no. 2, 407-421.
  18. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, 1989.
  19. A. Nachman and A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math. 38 (1980), no. 2, 275-282.
  20. S. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal. 3 (1979), no. 6, 897-904.
  21. E. Zeidler, Nonlinear Functional Analysis and its Applications. III, Springer-Verlag, 1985.

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