On the existence of multiple positive entire solutions for a class of quasilinear elliptic equations

Zuodong, Yang

doi:https://doi.org/10.1155/IJMMS/2006/34538

International Journal of Mathematics and Mathematical Sciences

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Volume 2006 | Article ID 034538 | https://doi.org/10.1155/IJMMS/2006/34538

On the existence of multiple positive entire solutions for a class of quasilinear elliptic equations

Yang Zuodong¹

Received21 Jun 2005

Revised22 Nov 2005

Accepted22 Jan 2006

Published13 Jul 2006

Abstract

Our goal is to establish the theorems of existence and multiple of positive entire solutions for a class quasilinear elliptic equations in ℝN with the Schauder-Tychonoff fixed point theorem as the principal tool. In many articles, the theorems of existence and multiple of positive entire solutions for a class semilinear elliptic equations are established. The results of the semilinear equations are extended to the quasilinear ones and the results of semilinear equations are developed.

References

H. Amann, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,” SIAM Review, vol. 18, no. 4, pp. 620–709, 1976.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. R. Esteban and J. L. Vázquez, “On the equation of turbulent filtration in one-dimensional porous media,” Nonlinear Analysis, vol. 10, no. 11, pp. 1303–1325, 1986.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
B. Gidas, W. M. Ni, and L. Nirenberg, “Symmetry and related properties via the maximum principle,” Communications in Mathematical Physics, vol. 68, no. 3, pp. 209–243, 1979.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Guedda and L. Véron, “Local and global properties of solutions of quasilinear elliptic equations,” Journal of Differential Equations, vol. 76, no. 1, pp. 159–189, 1988.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Z. M. Guo, “Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems,” Nonlinear Analysis, vol. 18, no. 10, pp. 957–971, 1992.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Z. M. Guo and J. R. L. Webb, “Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 124, no. 1, pp. 189–198, 1994.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. A. Herrero and J. L. Vázquez, “On the propagation properties of a nonlinear degenerate parabolic equation,” Communications in Partial Differential Equations, vol. 7, no. 12, pp. 1381–1402, 1982.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Kamin and L. Véron, “Flat core properties associated to the $p$ -Laplace operator,” Proceedings of the American Mathematical Society, vol. 118, no. 4, pp. 1079–1085, 1993.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Kichenassamy and J. Smoller, “On the existence of radial solutions of quasi-linear elliptic equations,” Nonlinearity, vol. 3, no. 3, pp. 677–694, 1990.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
T. Kusano and C. A. Swanson, “Positive entire solutions of semilinear biharmonic equations,” Hiroshima Mathematical Journal, vol. 17, no. 1, pp. 13–28, 1987.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
T. Kusano and H. Usami, “Positive solutions of a class of second order semilinear elliptic equations in the plane,” Mathematische Annalen, vol. 268, no. 2, pp. 255–264, 1984.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
View at: MathSciNet
G. M. Lieberman, “Boundary regularity for solutions of degenerate elliptic equations,” Nonlinear Analysis, vol. 12, no. 11, pp. 1203–1219, 1988.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Q. Lu, Z. Yang, and E. H. Twizell, “Existence of entire explosive positive solutions of quasi-linear elliptic equations,” Applied Mathematics and Computation, vol. 148, no. 2, pp. 359–372, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
M. Montenegro and M. Montenegro, “Existence and nonexistence of solutions for quasilinear elliptic equations,” Journal of Mathematical Analysis and Applications, vol. 245, no. 2, pp. 303–316, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. V. Pao, “Positive solutions of a nonlinear boundary-value problem of parabolic type,” Journal of Differential Equations, vol. 22, no. 1, pp. 145–163, 1976.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. I. Pohožaev, “On equations of the form $\vartriangle u = f(x,\,u,\,\nabla u)$ ,” Mathematics of the USSR. Sbornik, vol. 41, no. 2, pp. 269–280, 1982.
View at: Publisher Site | Google Scholar
D. Ruiz, “A priori estimates and existence of positive solutions for strongly nonlinear problems,” Journal of Differential Equations, vol. 199, no. 1, pp. 96–114, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
P. Tolksdorf, “On the Dirichlet problem for quasilinear equations in domains with conical boundary points,” Communications in Partial Differential Equations, vol. 8, no. 7, pp. 773–817, 1983.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
X. Xu, “Positive entire solutions to a class of nonlinear elliptic equations,” Journal of Mathematics (Wuhan), vol. 16, no. 4, pp. 403–411, 1996 (Chinese).
View at: Google Scholar | Zentralblatt MATH | MathSciNet
X. Xu, B. Yang, and L. Debnath, “Positive entire solutions of nonlinear polyharmonic equations in $\mathbb{R}^2$ ,” Applied Mathematics and Computation, vol. 126, no. 2-3, pp. 377–388, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Yang, “Non-existence of positive entire solutions for elliptic inequalities of $p$ -Laplacian,” Applied Mathematics. A Journal of Chinese Universities. Series B, vol. 12, no. 4, pp. 399–410, 1997 (Chinese).
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Z. Yang, “Existence of entire explosive positive radial solutions of quasilinear elliptic systems,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 46, pp. 2907–2927, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Yang, “Existence of positive bounded entire solutions for quasilinear elliptic equations,” Applied Mathematics and Computation, vol. 156, no. 3, pp. 743–754, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Yang, “Existence of positive entire solutions for singular and non-singular quasi-linear elliptic equation,” to appear in Journal of Computational and Applied Mathematics, Corrected Proof, Available online 4 January 2006.
View at: Google Scholar
Z. Yang and Z. Guo, “On the structure of positive solutions for quasilinear ordinary differential equations,” Applicable Analysis, vol. 58, no. 1-2, pp. 31–52, 1995.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Z. Yang and H. Yang, “Existence of positive radial solutions for quasilinear elliptic boundary value problems,” Annals of Differential Equations, vol. 15, no. 4, pp. 438–452, 1999 (Chinese).
View at: Google Scholar | Zentralblatt MATH | MathSciNet

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Copyright © 2006 Hindawi Publishing Corporation. 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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