Academia.eduAcademia.edu

Outline

The Thomas-Fermi method and polarizability of nuclei

1990, Nuclear Physics A

https://doi.org/10.1016/0375-9474(90)90299-2

Abstract

We establish several results related to existence, nonexistence or bifurcation of positive solutions for the boundary value problem −∆u + K(x)g(u) + |∇u| a = λf (x, u) in Ω, u = 0 on ∂Ω, where Ω ⊂ R N (N ≥ 2) is a smooth bounded domain, 0 < a ≤ 2, λ is a positive parameter, and f is smooth and has a sublinear growth. The main feature of this paper consists in the presence of the singular nonlinearity g combined with the convection term |∇u| a . Our approach takes into account both the sign of the potential K and the decay rate around the origin of the singular nonlinearity g. The proofs are based on various techniques related to the maximum principle for elliptic equations.

References (25)

  1. G. Barles, G. D íaz, and J. I. D íaz, Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a non lipschitz nonlinearity, Comm. Partial Differential Equations 17 (1992), 1037-1050.
  2. P. Bénilan, H. Brezis, and M. Crandall, A semilinear equation in L 1 (R N ), Ann. Scuola Norm. Sup. Pisa 4 (1975), 523-555.
  3. C. M. Brauner and B. Nicolaenko, On nonlinear eigenvalue problems which extend into free boundaries, Bifurcation and nonlinear eigenvalue problems (Proc., Session, Univ. Paris XIII, Villetaneuse, 1978), pp. 61-100, Lecture Notes in Math., 782, Springer, Berlin-New York, 1980.
  4. H. Brezis and S. Kamin, Sublinear elliptic equations in R N , Manuscripta Math. 74 (1992), 87-106.
  5. L. Caffarelli, R. Hardt, and L. Simon, Minimal surfaces with isolated singularities, Manuscripta Math. 48 (1984), 1-18.
  6. A. Callegari and A. Nashman, Some singular nonlinear equations arising in boundary layer theory, J. Math. Anal. Appl. 64 (1978), 96-105.
  7. A. Callegari and A. Nashman, A nonlinear singular boundary value problem in the theory of pseudo-plastic fluids, SIAM J. Appl. Math. 38 (1980), 275-281.
  8. F.-C. Cîrstea, M. Ghergu, and V. Rȃdulescu, Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type, J. Math. Pures Appl., in press.
  9. M. M. Coclite and G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations 14 (1989), 1315-1327.
  10. M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), 193-222.
  11. J. I. D íaz, Nonlinear partial differential equations and free boundaries. Vol. I. Elliptic equations Research Notes in Mathematics, 106. Pitman (Advanced Publishing Program), Boston, MA, 1985.
  12. J. I. D íaz, J. M. Morel, and L. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations 12 (1987), 1333-1344.
  13. W. Fulks and J. S. Maybee, A singular nonlinear equation, Osaka J. Math. 12 (1960), 1-19.
  14. M. Ghergu and V. Rȃdulescu, Sublinear singular elliptic problems with two parameters, J. Differential Equations 195 (2003), 520-536.
  15. M. Ghergu and V. Rȃdulescu, Bifurcation for a class of singular elliptic problems with quadratic convection term, C. R. Acad. Sci. Paris, Ser. I 338 (2004), 831-836.
  16. M. Ghergu and V. Rȃdulescu, Multiparameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with a convection term, Proc. Royal Soc. Edinburgh, Sect. A, in press.
  17. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
  18. Y. Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential Equations 189 (2003), 487-512.
  19. J. Hernández, F. J. Mancebo, and J. M. Vega, On the linearization of some singular nonlinear elliptic problems and applications, Ann. Inst. H. Poincaré, Anal. Non Linéaire 19 (2002), 777-813.
  20. J. Hernández, F. J. Mancebo, and J. M. Vega, Nonlinear singular elliptic problems: recent results and open problems, Preprint, 2005.
  21. A. Meadows, Stable and singular solutions of the equation ∆u = 1/u, Indiana Univ. Math. J. 53 (2004), 1681-1703.
  22. J. Shi and M. Yao, On a singular nonlinear semilinear elliptic problem, Proc. Royal Soc. Edinburgh, Sect. A 128 (1998), 1389-1401.
  23. J. Shi and M. Yao, Positive solutions for elliptic equations with singular nonlinearity, Electronic Journal of Differential Equations 4 (2005), 1-11.
  24. C. A. Stuart, Existence and approximation of solutions of nonlinear elliptic equations, Math. Z. 147 (1976), 53-63.
  25. Z. Zhang, Nonexistence of positive classical solutions of a singular nonlinear Dirichlet problem with a convection term, Nonlinear Anal., T.M.A. 8 (1996), 957-961.