Abstract

We provide a variational approach to singular semilinear elliptic equations of the form −Δu=u−β+g(x,u)-\Delta u=u^{-\beta}+g(x,u), for every β>0\beta > 0.

Author information

Contact details are reproduced from the original publication and may be historical.

Annamaria Canino

Dip. di Matematica, Università della Calabria, 87036 Arcavacata di Rende, Italy

canino@unical.it

Marco Degiovanni

Dip. di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41,
25121 Brescia, Italy

m.degiovanni@dmf.unicatt.it

Keywords

  • Singular semilinear elliptic equations
  • variational methods
  • convex functionals

Mathematics Subject Classification

35J65, 49J40

References

  1. H. Attouch: Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman Advanced Publishing Program, Boston, London (1984).Google Scholar
  2. H. Brezis: Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in: Contrib. Nonlin. Functional Analysis, Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, 1971, Academic Press, New York (1971) 101–156.https://doi.org/10.1016/B978-0-12-775850-3.50009-1Google Scholar
  3. H. Brezis, F. E. Browder: Sur une propriété des espaces de Sobolev, C. R. Acad. Sci., Paris, Sér. A 287 (1978) 113–115.Google Scholar
  4. H. Brezis, T. Kato: Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. 58(9) (1979) 137–151.Google Scholar
  5. Y. S. Choi, A. C. Lazer, P. J. McKenna: Some remarks on a singular elliptic boundary value problem, Nonlinear Anal. 32 (1998) 305–314.https://doi.org/10.1016/S0362-546X(97)00492-6Google Scholar
  6. M. G. Crandall, P. H. Rabinowitz, L. Tartar: On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differ. Equations 2 (1977) 193–222.https://doi.org/10.1080/03605307708820029Google Scholar
  7. G. Dal Maso: An Introduction to Γ-Convergence, Progress in Nonlinear Differential Equations and their Applications 8, Birkhäuser, Boston (1993).https://doi.org/10.1007/978-1-4612-0327-8Google Scholar
  8. E. De Giorgi, T. Franzoni: Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58(8) (1975) 842–850.Google Scholar
  9. J. I. Diaz, J.-M. Morel, L. Oswald: An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations 12 (1987) 1333–1344.https://doi.org/10.1080/03605308708820531Google Scholar
  10. J. A. Gatica, V. Oliker, P. Waltman: Singular nonlinear boundary value problems for second-order ordinary differential equations, J. Differ. Equations 79 (1989) 62–78.https://doi.org/10.1016/0022-0396(89)90113-7Google Scholar
  11. L. L. Helms: Introduction to Potential Theory, Pure and Applied Mathematics 22, Wiley-Interscience, New York, London, Sydney (1969).Google Scholar
  12. N. Hirano, C. Saccon, N. Shioji: Multiple existence of positive solutions for singular elliptic problems with concave and convex nonlinearities, preprint (2003).https://doi.org/10.57262/ade/1355867973Google Scholar
  13. B. Kawohl: On a class of singular elliptic equations, in: Progress in Partial Differential Equations: Elliptic and Parabolic Problems, C. Bandle et al. (ed.), Pont-à-Mousson, 1991, Pitman Res. Notes Math. Ser. 266, Longman Sci. Tech., Harlow (1992) 156–163.Google Scholar
  14. D. Kinderlehrer, G. Stampacchia: An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics 88, Academic Press, New York, London, Toronto (1980).Google Scholar
  15. A. V. Lair, A. W. Shaker: Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl. 211 (1997) 371–385.https://doi.org/10.1006/jmaa.1997.5470Google Scholar
  16. A. C. Lazer, P. J. McKenna: On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc. 111 (1991) 721–730.https://doi.org/10.1090/S0002-9939-1991-1037213-9Google Scholar
  17. U. Mosco: Convergence of convex sets and of solutions of variational inequalities, Adv. in Math. 3 (1969) 510–585.https://doi.org/10.1016/0001-8708(69)90009-7Google Scholar
  18. C. A. Stuart: Existence and approximation of solutions of non-linear elliptic equations, Math. Z. 147 (1976) 53–63.https://doi.org/10.1007/BF01214274Google Scholar
  19. A. Szulkin: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986) 77–109.https://doi.org/10.1016/s0294-1449(16)30389-4Google Scholar
  20. N. Zeddini: Positive solutions of singular elliptic equations outside the unit disk, Electron. J. Differ. Equ. 2001, Paper No. 53 (2001) (electronic only).Google Scholar

Suggested citation

A. Canino, M. Degiovanni. “A Variational Approach to a Class of Singular Semilinear Elliptic Equations.” Journal of Convex Analysis 11 (2004), No. 1, 147–162. https://doi.org/10.68381/jca11010

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Published by Heldermann Verlag, 2004. Rights now held by Banach Press.