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Toumi, Faten

doi:https://doi.org/10.1155/AAA/2006/95480

Abstract and Applied Analysis

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Volume 2006 | Article ID 095480 | https://doi.org/10.1155/AAA/2006/95480

Existence of positive solutions for nonlinear boundary value problems in bounded domains of ℝn

Faten Toumi¹

Received10 Jun 2004

Accepted22 Sept 2004

Published21 Feb 2006

Abstract

Let D be a bounded domain in ℝn(n≥2). We consider the following nonlinear elliptic problem: Δu=f(⋅,u) in D (in the sense of distributions), u|∂D=ϕ, where ϕ is a nonnegative continuous function on ∂D and f is a nonnegative function satisfying some appropriate conditions related to some Kato class of functions K(D). Our aim is to prove that the above problem has a continuous positive solution bounded below by a fixed harmonic function, which is continuous on D¯. Next, we will be interested in the Dirichlet problem Δu=−ρ(⋅,u) in D (in the sense of distributions), u|∂D=0, where ρ is a nonnegative function satisfying some assumptions detailed below. Our approach is based on the Schauder fixed-point theorem.

References

M. Aizenman and B. Simon, “Brownian motion and Harnack inequality for Schrödinger operators,” Communications on Pure and Applied Mathematics, vol. 35, no. 2, pp. 209–273, 1982.
View at: Google Scholar
S. Athreya, “On a singular semilinear elliptic boundary value problem and the boundary Harnack principle,” Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis, vol. 17, no. 3, pp. 293–301, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Brezis and S. Kamin, “Sublinear elliptic equations in ${\mathbf R}\sp n$ ,” 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. Dalmasso, “Existence and uniqueness results for polyharmonic equations,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 36, no. 1, pp. 131–137, 1999.
View at: Google Scholar
R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 1, L'oprateur de Laplace, INSTN: Collection Enseignement. [INSTN: Teaching Collection], Masson, Paris, 1987.
N. J. Kalton and I. E. Verbitsky, “Nonlinear equations and weighted norm inequalities,” Transactions of the American Mathematical Society, vol. 351, no. 9, pp. 3441–3497, 1999.
View at: Google Scholar
H. Mâagli, “Inequalities for the Riesz potentials,” Archives of Inequalities and Applications. An International Journal for Theory and Applications, vol. 1, no. 3-4, pp. 285–294, 2003.
View at: Google Scholar
H. Mâagli and L. Mâatoug, “Singular solutions of a nonlinear equation in bounded domains of $\mathbf{R}\sp 2$ ,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 230–246, 2002.
View at: Google Scholar
H. Mâagli and M. Zribi, “On a new Kato class and singular solutions of a nonlinear elliptic equation in bounded domains of $\mathbb{R}^{n}$ ,” to appear in Positivity.
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.
M. Selmi, “Inequalities for Green functions in a Dini-Jordan domain in $\mathbf{R}\sp 2$ ,” Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis, vol. 13, no. 1, pp. 81–102, 2000.
View at: Google Scholar
N. Zeddini, “Positive solutions for a singular nonlinear problem on a bounded domain in $\mathbb R\sp 2$ ,” Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis, vol. 18, no. 2, pp. 97–118, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Z. X. Zhao, “Uniform boundedness of conditional gauge and Schrödinger equations,” Communications in Mathematical Physics, vol. 93, no. 1, pp. 19–31, 1984.
View at: Google Scholar

Copyright

Copyright © 2006 Faten Toumi. 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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