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Positive solutions of some quasilinear singular second order equations

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  09 April 2009

J. V. Goncalves  and
C. A. P. Santos
J. V. Goncalves
Affiliation:
Universidade de Brasilia, Departamento de Matemática, 70910-900 Brasilia(DF), Brazil, e-mail: jv@mat.unb.br
C. A. P. Santos
Affiliation:
Universidade Federal de Goiás, Departmento de Matemática Catalão(GO), Brazil, e-mail: csantos@unb.br
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Abstract

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In this paper we study the existence and uniqueness of positive solutions of boundary vlue problems for continuous semilinear perturbations, say f: [0, 1) × (0, ∞) → (0, ∞), of class of quasilinear operators which represent, for instance, the radial form of the Dirichlet problem on the unit ball of RN for the operators: p-Laplacian (1 < p < ∞) ad k-Hessian (1 ≤ k ≤ N). As a key feature, f (r, u) is possibly singular at r = 1 or u =0, Our approach exploits fixed point arguments and the Shooting Method.

Keywords

quasilinear singular eqationsraidal positive solutionsfixed pointsshooting mehtod

MSC classification

Secondary: 35J25: Boundary value problems for second-order elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 1 , February 2004 , pp. 125 - 140
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Brézis, H. and Oswald, L., ‘Remarks on sublinear elliptic equations’, Nonlinear Anal. 10 (1986), 55–64.CrossRefGoogle Scholar
[2]Callegari, A. and Nachman, A., ‘A nonlinear singular boundary value problem in the theory of pseudoplastic fluids’, SIAM J. Appl. Math. 38 (1980), 275–281.Google Scholar
[3]Chabrowski, J., ‘Existence results for singular elliptic equations’, Hokkaido Math. J. 20 (1991), 465–475.CrossRefGoogle Scholar
[4]Choi, Y. S. and Kim, E. H., ‘On the existence of positive solutions of quasilinear ellipitic boundary value problems’, J. Differential Equations 155 (1999), 423–442.CrossRefGoogle Scholar
[5]Choi, Y. S., Lazer, A. C. and McKenna, P. J., ‘On a singular quasilinear anisotropic elliptic boundary value problem’, Trans. Amer. Math. Soc. 347 (1995), 2633–2641.CrossRefGoogle Scholar
[6]Clement, Ph., Figueiredo, D. G. and Mitidieri, E., ‘Quasilinear elliptic equations with critical exponents’, Topol. Methods Nonlinear Anal. 7 (1996), 133–170.CrossRefGoogle Scholar
[7]Clement, Ph., Manasevich, R. and Mitidieri, E., ‘Some existence and nonexistence results for a homogeneous quasilinear problem’, Asymptot. Anal. 17 (1998), 13–29.Google Scholar
[8]Crandall, M., Rabinowitz, P. and Tartar, L., ‘On a Dirichlet problem with singular nonlinearity’, Comm. Partial Differential Equations 2 (1977), 193–222.CrossRefGoogle Scholar
[9]Diaz, J. I. and Saa, J. E., ‘Existence et unicité de solutions positives pour certaines equations elliptiques quasilinéaires’, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 521–524.Google Scholar
[10]Figueiredo, D. G.Goncalves, J. V. and Miyagaki, O. H., ‘On a class of quasilinear elliptic problems involving critical Sobolev exponents’, Commun. Contemp. Math. 2 (2000), 47–59.CrossRefGoogle Scholar
[11]Fulks, W. and Maybee, J. S., ‘A singular nonlinear equation’, Osaka Math. J. 12 (1960), 1–19.Google Scholar
[12]Hai, D. D. and Oppenheimer, S. F., ‘Singular boundary value problems for p-Laplacian-like equations’, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 975–981.CrossRefGoogle Scholar
[13]Hai, D. D., Schmitt, K. and Shivaji, R., ‘On the number of solutions of boundary value problems involving the p-Laplacian’, Electron. J. Differential Equations 1996 (1996), 1–9.Google Scholar
[14]Hai, D. D., Schmitt, K. and Shivaji, R., ‘Positive solutions of quasilinear boundary value problems’, J. Math. Anal. Appl. 217 (1998), 672–686.CrossRefGoogle Scholar
[15]Lair, A. V. and Shaker, A., ‘Classical and weak solutions of a singular semilinear elliptic problem’, J. Math. Anal. Appl. 211 (1997), 371–385.CrossRefGoogle Scholar
[16]Simon, J., Regularité de la solution d'une equation non lineaire dans RN Lecture Notes in Math. 665 (Springer, Berlin, 1978) pp. 203–227.Google Scholar
[17]Swanson, C. A. and Kusano, T., ‘Entire positive solutions of singular semilinear elliptic equations’, Japan J. Math. 11 (1985), 145–155.Google Scholar
[18]Taliaferro, S. D., ‘A nonlinear singular boundary value problem’, Nonlinear Anal. 3 (1979), 897–894.CrossRefGoogle Scholar
[19]Tso, K., ‘On a real monge-ampère functional’, Invent. Math. 101 (1990), 425–448.CrossRefGoogle Scholar
[20]Tso, K., ‘Remarks on critical exponents for Hessian operators’, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 113–122.CrossRefGoogle Scholar
[21]Wong, F. H., ‘Uniqueness of positive solutions for Sturm-Liouville boundary value problems’, Proc. Amer. Math. Soc. 126 (1998), 365–374.CrossRefGoogle Scholar
[22]Wong, F. H., ‘Existence of positive solutions for m-Laplacian boundary value problems’, Appl. Math. Lett. 12 (1999), 11–17.CrossRefGoogle Scholar
[23]Zhang, Z., ‘On a Dirichlet problem with singular nonlinearity’, J. Math. Anal. Appl. 194 (1995), 103–113.CrossRefGoogle Scholar