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

Mâagli, Habib; Zribi, Malek

doi:https://doi.org/10.1155/AAA/2006/76582

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Volume 2006 | Article ID 076582 | https://doi.org/10.1155/AAA/2006/76582

Existence of positive solutions for some polyharmonic nonlinear equations in ℝn

Habib Mâagli¹and Malek Zribi¹

Received25 Dec 2004

Accepted01 Mar 2005

Published23 Feb 2006

Abstract

We will study the following polyharmonic nonlinear elliptic equation (−∆)mu+f(‧,u)=0 in ℝn, n>2m. Under appropriate conditions on the nonlinearity f(x,t), related to a class of functions called m-Green-tight functions, we give some existence results for the above equation.

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
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: Google Scholar
H. Brezis and S. Kamin, “Sublinear elliptic equations in $\mathbf{R}^n$ ,” 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. Dalmasso, “Existence and uniqueness results for polyharmonic equations,” Nonlinear Analysis, Ser. A: Theory Methods, vol. 36, no. 1, pp. 131–137, 1999.
View at: Google Scholar
H.-C. Grunau and G. Sweers, “The maximum principle and positive principal eigenfunctions for polyharmonic equations,” in Reaction Diffusion Systems (Trieste, 1995), G. Caristi and E. Mitidieri, Eds., vol. 194 of Lecture Notes in Pure and Appl. Math., pp. 163–182, Dekker, New York, 1998.
View at: Google Scholar
H. Mâagli and S. Masmoudi, “Existence and asymptotic behaviour of large solutions of semilinear elliptic equations,” Potential Analysis, vol. 17, no. 4, pp. 337–350, 2002.
View at: Google Scholar
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
J. Trubek, “Asymptotic behavior of solutions to $\Delta u + Ku^\sigma=0$ on $\mathbf{R}^n$ for $n \geqslant 3$ ,” Proceedings of the American Mathematical Society, vol. 106, no. 4, pp. 953–959, 1989.
View at: Google Scholar
Z. Yin, “Bounded positive solutions of Schrödinger equations,” Applied Mathematics Letters, vol. 17, no. 5, pp. 527–533, 2004.
View at: Google Scholar
Q. S. Zhang and Z. X. Zhao, “Singular solutions of semilinear elliptic and parabolic equations,” Mathematische Annalen, vol. 310, no. 4, pp. 777–794, 1998.
View at: Google Scholar
Z. X. Zhao, “Green function for Schrödinger operator and conditioned Feynman-Kac gauge,” Journal of Mathematical Analysis and Applications, vol. 116, no. 2, pp. 309–334, 1986.
View at: Google Scholar
Z. X. Zhao, “On the existence of positive solutions of nonlinear elliptic equations—a probabilistic potential theory approach,” Duke Mathematical Journal, vol. 69, no. 2, pp. 247–258, 1993.
View at: Google Scholar

Copyright

Copyright © 2006 Habib Mâagli and Malek Zribi. 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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