Hostname: page-component-7bb8b95d7b-s9k8s Total loading time: 0 Render date: 2024-09-13T22:23:20.185Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence of positive radial solutions for a class of nonlinear singular elliptic problems in annular domains

Published online by Cambridge University Press:  20 January 2009

Zongming Guo
Zongming Guo
Affiliation:
Department of MathematicsUniversity of GlasgowGlasgow G12 8QWScotland Henan Normal UniversityXinxiangP.R.China
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish the existence of positive radially symmetric solutions of Δu + f(r, u, u′) = 0 in the domain R1 <r<R0 with a variety of Dirichlet and Neumann boundary conditions. The function f is allowed to be singular when either u = 0 or u′ = 0. Our analysis is based on Leray-Schauder degree theory.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 35 , Issue 3 , October 1992 , pp. 405 - 418
Copyright
Copyright © Edinburgh Mathematical Society 1992

References

REFERENCES

1.Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.CrossRefGoogle Scholar
2.Bandle, C., Coffman, C. V. and Marcus, M., Nonlinear elliptic problems in annular domains, J. Differential Equations 69 (1987), 322–345.CrossRefGoogle Scholar
3.Bobisud, L. E., O'Regan, D. and Royalty, W. D., Solvability of some nonlinear boundary value problems, Nonlinear Analysis, T. M. A. 12 (9) (1988), 855–869.CrossRefGoogle Scholar
4.Brezis, H. and Turner, E. L., On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977), 601–614.CrossRefGoogle Scholar
5.Coffman, C. V., On the positive solutions of boundary value problems for a class of nonlinear differential equations, J. Differential Equations 3 (1967), 92–111.CrossRefGoogle Scholar
6.Garaizer, X., Existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differential Equations 70 (1987), 69–92.CrossRefGoogle Scholar
7.Gatica, J. A., Oliker, V. and Waltman, P., Singular nonlinear boundary value problems for second order ordinary differential equations, J. Differential Equations 79 (1989), 62–78.CrossRefGoogle Scholar
8.Gelfand, I. M., Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. 1 (2) 29 (1963), 295–381.Google Scholar
9.Gidas, B., Ni, W. M. and Nirenberg, L., Symmetry and related properties in almost spherically symmetric domains, Arch. Rat. Mech. Anal. 96 (1986), 167–197.Google Scholar
10.Gidas, B., Ni, W. M. and Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243.CrossRefGoogle Scholar
11.Guo, Z. M., Solvability of some singular nonlinear boundary value problems and existence of positive radial solutions of some nonlinear elliptic problems, Nonlinear Analysis 16 (1991), 781–790.CrossRefGoogle Scholar
12.Lin, S. S., On the existence of positive radial solutions for nonlinear elliptic equations in annular domains, J. Differential Equations 81 (1989), 221–223.CrossRefGoogle Scholar
13.Lin, S. S., Positive radial solutions and non-radial bifurcation for semilinear elliptic equations in annulus domains, J. Differential Equations 86 (1990), 367–391.CrossRefGoogle Scholar
14.Lions, P. L., On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), 441–467.CrossRefGoogle Scholar
15.Ni, W. M., Uniqueness of solutions of nonlinear Dirichlet problems. J. Differential Equations 50 (1983), 289–304.CrossRefGoogle Scholar
16.Ni, W. M. and Nussbaum, R., Uniqueness and non-uniqueness for positive radial solutions of Δu + f(u, r) = 0, Comm. Pure Appl. Math. 38 (1985), 67–108.CrossRefGoogle Scholar
17.O'Regan, D., Positive solutions to singular and non-singular second-order boundary value problems, J. Math. Anal. Appl. 142 (1989), 40–52.CrossRefGoogle Scholar
18.Pohozaev, S. I., Eigenfunction of the equation Δu + λf(u) = 0, Soviet Math. Dokl. 5 (1965), 1408–1411.Google Scholar