Existence of positive solutions for some nonlinear
elliptic problems in unbounded domains of <mml:math alttext="${\mathbb{R}}^n$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>ℝ</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math>

Zeddini, Noureddine

doi:https://doi.org/10.1155/AAA/2006/24307

Abstract and Applied Analysis

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Volume 2006 | Article ID 024307 | https://doi.org/10.1155/AAA/2006/24307

Existence of positive solutions for some nonlinear elliptic problems in unbounded domains of ℝn

Noureddine Zeddini¹

Received26 May 2005

Accepted01 Aug 2005

Published14 Mar 2006

Abstract

This paper deals with a class of nonlinear elliptic equations in an unbounded domain D of ℝn, n≥3, with a nonempty compact boundary, where the nonlinear term satisfies some appropriate conditions related to a certain Kato class K∞(D). Our purpose is to give some existence results and asymptotic behaviour for positive solutions by using the Green function approach and the Schauder fixed point theorem.

References

I. Bachar, H. Mâagli, and N. Zeddini, “Estimates on the Green function and existence of positive solutions of nonlinear singular elliptic equations,” Communications in Contemporary Mathematics, vol. 5, no. 3, pp. 401–434, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Brezis and S. Kamin, “Sublinear elliptic equations in $\mathbb{R}^n$ ,” Manuscripta Mathematica, vol. 74, no. 1, pp. 87–106, 1992.
View at: Google Scholar
K. L. Chung and Z. X. Zhao, From Brownian Motion to Schrödinger's Equation, vol. 312 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, 1995.
R. Dautray and J.-L. Lions, “Analyse mathématique et calcul numérique pour les sciences et les techniques,” in L' opérateur de Laplace, vol. 2 of Coll. C.E.A, Masson, Paris, 1987.
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
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. Shaker, “Classical and weak solutions of a singular semilinear elliptic problem,” Journal of Mathematical Analysis and Applications, vol. 211, no. 2, pp. 371–385, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | 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, “Perturbation semi-linéaire des résolvantes et des semi-groupes [Semilinear perturbation of resolvents and semigroups],” Potential Analysis, vol. 3, no. 1, pp. 61–87, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Mâagli and S. Masmoudi, “Positive solutions of some nonlinear elliptic problems in unbounded domain,” Annales Academiæ Scientiarium Fennicæ . Mathematica, vol. 29, no. 1, pp. 151–166, 2004.
View at: Google Scholar
H. Mâagli and M. Zribi, “Existence and estimates of solutions for singular nonlinear elliptic problems,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 522–542, 2001.
View at: Google Scholar
S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory. Probability and Mathematical Statistics, Academic Press, New York, 1978.

Copyright

Copyright © 2006 Noureddine Zeddini. 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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