On a singular sublinear polyharmonic problem

Ben Othman, Sonia

doi:https://doi.org/10.1155/AAA/2006/27969

Abstract and Applied Analysis

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Volume 2006 | Article ID 027969 | https://doi.org/10.1155/AAA/2006/27969

On a singular sublinear polyharmonic problem

Sonia Ben Othman¹

Received27 Jun 2005

Accepted05 Sept 2005

Published31 May 2006

Abstract

This paper deals with a class of singular nonlinear polyharmonic equations on the unit ball B in ℝn (n≥2) where the combined effects of a singular and a sublinear term allow us by using the Schauder fixed point theorem to establish an existence result for the following problem: (−Δ)mu=φ(⋅,u)+ψ(⋅,u) in B (in the sense of distributions), u>0, lim⁡|x|→1u(x)/(1−|x|)m−1=0. Our approach is based on estimates for the polyharmonic Green function on B with zero Dirichlet boundary conditions.

References

I. Bachar, H. Mâagli, S. Masmoudi, and M. Zribi, “Estimates for the Green function and singular solutions for polyharmonic nonlinear equation,” Abstract and Applied Analysis, vol. 2003, no. 12, pp. 715–741, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
T. Boggio, “Sulle funzioni di Green d'ordine $m$ ,” Rendiconti del Circolo Matematico di Palermo, vol. 20, pp. 97–135, 1905.
View at: Google Scholar
M. G. Crandall, P. H. Rabinowitz, and L. Tartar, “On a Dirichlet problem with a singular nonlinearity,” Communications in Partial Differential Equations, vol. 2, no. 2, pp. 193–222, 1977.
View at: Google Scholar
J. I. Diaz, J.-M. Morel, and L. Oswald, “An elliptic equation with singular nonlinearity,” Communications in Partial Differential Equations, vol. 12, no. 12, pp. 1333–1344, 1987.
View at: Google Scholar
A. L. Edelson, “Entire solutions of singular elliptic equations,” Journal of Mathematical Analysis and Applications, vol. 139, no. 2, pp. 523–532, 1989.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. R. Garabedian, “A partial differential equation arising in conformal mapping,” Pacific Journal of Mathematics, vol. 1, no. 4, pp. 485–524, 1951.
View at: Google Scholar
T. Kusano and C. A. Swanson, “Entire positive solutions of singular semilinear elliptic equations,” Japanese Journal of Mathematics. New Series, vol. 11, no. 1, pp. 145–155, 1985.
View at: Google Scholar
A. V. Lair and A. W. Wood, “Large solutions of sublinear elliptic equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 39, no. 6, pp. 745–753, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
A. C. Lazer and P. J. McKenna, “On a singular nonlinear elliptic boundary-value problem,” Proceedings of the American Mathematical Society, vol. 111, no. 3, pp. 721–730, 1991.
View at: Google Scholar
H. Mâagli, F. Toumi, and M. Zribi, “Existence of positive solutions for some polyharmonic nonlinear boundary-value problems,” Electronic Journal of Differential Equations, vol. 2003, no. 58, pp. 1–19, 2003.
View at: Google Scholar
J.-P. Shi and M. Yao, “On a singular nonlinear semilinear elliptic problem,” Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, vol. 128, no. 6, pp. 1389–1401, 1998.
View at: Google Scholar

Copyright

Copyright © 2006 Sonia Ben Othman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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