Duke Mathematical Journal

Boundary singularities of solutions of some nonlinear elliptic equations

Abdelilah Gmira and Laurent Véron

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Source
Duke Math. J., Volume 64, Number 2 (1991), 271-324.

Dates
First available in Project Euclid: 20 February 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1077295523

Digital Object Identifier
doi:10.1215/S0012-7094-91-06414-8

Mathematical Reviews number (MathSciNet)
MR1136377

Zentralblatt MATH identifier
0766.35015

Subjects
Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35A20: Analytic methods, singularities 35B05: Oscillation, zeros of solutions, mean value theorems, etc.

Citation

Gmira, Abdelilah; Véron, Laurent. Boundary singularities of solutions of some nonlinear elliptic equations. Duke Math. J. 64 (1991), no. 2, 271--324. doi:10.1215/S0012-7094-91-06414-8. https://projecteuclid.org/euclid.dmj/1077295523


Export citation

References

  • [1] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727.
  • [2] N. Aronszajn and K. T. Smith, Functional spaces and functional completion, Ann. Inst. Fourier. Grenoble 6 (1955–1956), 125–185.
  • [3] P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 1, 185–206.
  • [4] P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints, Ark. Mat. 22 (1984), no. 2, 153–173.
  • [5] Ph. Bénilan and H. Brézis, Nonlinear problems related to the Thomas-Fermi equation, preprint. (See [10].).
  • [6] P. Benilan, H. Brezis, and M. Crandall, A semilinear equation in $L\sp1(R\spN)$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 4, 523–555.
  • [7] H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. Funct. Anal. 40 (1981), no. 1, 1–29.
  • [8] M. Berger, P. Gauduchon, and E. Mazet, Le spectre d'une variété riemannienne, Lecture Notes in Mathematics, vol. 194, Springer-Verlag, Berlin, 1971.
  • [9] H. Brézis, Une équation semi-linéaire avec conditions aux limites dans $L^1$, preprint.
  • [10] H. Brezis, Some variational problems of the Thomas-Fermi type, Variational inequalities and complementarity problems (Proc. Internat. School, Erice, 1978) eds. R. W. Cottle, F. Giannessi, and J.-L. Lions, Wiley, Chichester, 1980, pp. 53–73.
  • [11] H. Brézis and E. H. Lieb, Long range atomic potentials in Thomas-Fermi theory, Comm. Math. Phys. 65 (1979), no. 3, 231–246.
  • [12] H. Brézis and L. Véron, Removable singularities for some nonlinear elliptic equations, Arch. Rational Mech. Anal. 75 (1980/81), no. 1, 1–6.
  • [13] E. Cartan, Complément au mémoire “Sur la géometrie des groupes simples”, Annali di Mat. 5 (1928), 253–260.
  • [14] X.-Y. Chen, H. Matano, and L. Veron, Anisotropic singularities of solutions of nonlinear elliptic equations in $\bf R\sp 2$, J. Funct. Anal. 83 (1989), no. 1, 50–97.
  • [15] M. Cotlar and R. Cignoli, An introduction to functional analysis, North-Holland Publishing Co., Amsterdam, 1974.
  • [16] M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340.
  • [17] R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 1, Collection du Commissariat à l'Énergie Atomique: Série Scientifique. [Collection of the Atomic Energy Commission: Science Series], Masson, Paris, 1984.
  • [18] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1977.
  • [19] A. Gmira, Comportement asymptotique et singularités des solutions de problémes quasilinéaires, Thèse de Doctorat d'Etat es-Sciences, University of Tours, 1989.
  • [20] R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math. 7 (1957), 1641–1647.
  • [21] P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1986.
  • [22] P. Rabinowitz, Variational methods for nonlinear eigenvalue problems, Eigenvalues Of Nonlinear Problems ed. G. Prodi, Cremonese, Rome, 1975, pp. 141–195.
  • [23] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302.
  • [24] J. Serrin, Isolated singularities of solutions of quasi-linear equations, Acta Math. 113 (1965), 219–240.
  • [25] E. M. Stein, Boundary behavior of holomorphic functions of several complex variables, vol. 9, Princeton University Press, Princeton, N.J., 1972.
  • [26] J. L. Vazquez and L. Véron, Singularities of elliptic equations with an exponential nonlinearity, Math. Ann. 269 (1984), no. 1, 119–135.
  • [27] J. L. Vazquez and L. Véron, Isolated singularities of some semilinear elliptic equations, J. Differential Equations 60 (1985), no. 3, 301–321.
  • [28] L. Véron, Singularités éliminables d'équations elliptiques non linéaires, J. Differential Equations 41 (1981), no. 1, 87–95.
  • [29] L. Véron, Global behaviour and symmetry properties of singular solutions of nonlinear elliptic equations, Ann. Fac. Sci. Toulouse Math. (5) 6 (1984), no. 1, 1–31.
  • [30] L. Véron, Geometric invariance of singular solutions of some nonlinear partial differential equations, Indiana Univ. Math. J. 38 (1989), no. 1, 75–100.
  • [31] L. Véron, Singular solutions of some nonlinear elliptic equations, Nonlinear Anal. 5 (1981), no. 3, 225–242.