Existence and Nonexistence of Entire Positive Solutions for a Class of Singular p-Laplacian Elliptic System

ABSTRACT

In this paper, we show the existence and nonexistence of entire positive solutions for a class of singular elliptic system We have that entire large positive solutions fail to exist if f and g are sublinear and b and d have fast decay at infinity. However, if f and g satisfy some growth conditions at infinity, and b, d are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.

In this paper, we show the existence and nonexistence of entire positive solutions for a class of singular elliptic system We have that entire large positive solutions fail to exist if f and g are sublinear and b and d have fast decay at infinity. However, if f and g satisfy some growth conditions at infinity, and b, d are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.

KEYWORDS

Singular p-Laplacian Elliptic System, Entire Positive Solution, Large Solution, Bounded Solution, Entire Large Positive Radial Solution

Singular p-Laplacian Elliptic System, Entire Positive Solution, Large Solution, Bounded Solution, Entire Large Positive Radial Solution

Cite this paper

nullD. Lei and Z. Yang, "Existence and Nonexistence of Entire Positive Solutions for a Class of Singular p-Laplacian Elliptic System,"*Advances in Pure Mathematics*, Vol. 1 No. 6, 2011, pp. 351-358. doi: 10.4236/apm.2011.16063.

nullD. Lei and Z. Yang, "Existence and Nonexistence of Entire Positive Solutions for a Class of Singular p-Laplacian Elliptic System,"

References

[1] S. Chen and G. Lu, “Existence and Nonexistence of Positive Solutions for a Class of Semilinear Elliptic Systems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 38, No. 7, 1999, pp. 919-932. doi:10.1016/S0362-546X(98)00143-6

[2] F. Cìrstea and V. R?dulescu, “Entire Solutions Blowing up at Infinity for Semilinear Elliptic Systems,” Journal de Mathématiques Pures et Appliquées, Vol. 81, No. 9, 2002, pp. 827-846. doi:10.1016/S0021-7824(02)01265-5

[3] C. Yarur, “Existence of Continuous and Singular Ground States for Semilinear Elliptic Systems,” Electronic Journal of Differential Equations, Vol. 1, 1998, pp. 1-27.

[4] R. Dalmasso, “Existence and Uniqueness of Positive Solutions of Semilinear Elliptic Systems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 39, No. 5, 2000, pp. 559-568. doi:10.1016/S0362-546X(98)00221-1

[5] A. V. Lair and A. W. Wood, “Existence of Entire Large Positive Solutions of Semilinear Elliptic Systems,” Journal of Differential Equations, Vol. 164, No. 2, 2000, pp. 380-394. doi:10.1006/jdeq.2000.3768

[6] Z. D. Yang, “Existence of Entire Explosive Positive Radial Solutions for a Class of Quasilinear Elliptic Systems,” Journal of Mathematical Analysis and Applications, Vol. 288, No. 2, 2003, pp. 768-783. doi:10.1016/j.jmaa.2003.09.027

[7] H. X. Qing and Z. D. Yang, “A Remark on the Existence of Entire Positive Solutions for a Class of Quasilinear Elliptic Problem,” Oriental Journal of Mathematics, Vol. 3, No. 1, 2010, pp. 43-50.

[8] C. S. Chen, Z. Q. Wang and F. P. Wang,” Existence and Nonexistence of Positive Solutions for Singular p-Laplacian Equation in Rn,” Boundary Value Problems, Vol. 2010, 2010, Article ID 607453, pp. 1-17.

[9] X. G. Zhang and L. S. Liu, “The Existence and Nonexistence of Entire Positive Solutions of Semilinear Elliptic Systems with Gradient Term,” Journal of Mathematical Analysis and Applications, Vol. 371, No. 1, 2010, pp. 300-308. doi:10.1016/j.jmaa.2010.05.029

[1] S. Chen and G. Lu, “Existence and Nonexistence of Positive Solutions for a Class of Semilinear Elliptic Systems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 38, No. 7, 1999, pp. 919-932. doi:10.1016/S0362-546X(98)00143-6

[2] F. Cìrstea and V. R?dulescu, “Entire Solutions Blowing up at Infinity for Semilinear Elliptic Systems,” Journal de Mathématiques Pures et Appliquées, Vol. 81, No. 9, 2002, pp. 827-846. doi:10.1016/S0021-7824(02)01265-5

[3] C. Yarur, “Existence of Continuous and Singular Ground States for Semilinear Elliptic Systems,” Electronic Journal of Differential Equations, Vol. 1, 1998, pp. 1-27.

[4] R. Dalmasso, “Existence and Uniqueness of Positive Solutions of Semilinear Elliptic Systems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 39, No. 5, 2000, pp. 559-568. doi:10.1016/S0362-546X(98)00221-1

[5] A. V. Lair and A. W. Wood, “Existence of Entire Large Positive Solutions of Semilinear Elliptic Systems,” Journal of Differential Equations, Vol. 164, No. 2, 2000, pp. 380-394. doi:10.1006/jdeq.2000.3768

[6] Z. D. Yang, “Existence of Entire Explosive Positive Radial Solutions for a Class of Quasilinear Elliptic Systems,” Journal of Mathematical Analysis and Applications, Vol. 288, No. 2, 2003, pp. 768-783. doi:10.1016/j.jmaa.2003.09.027

[7] H. X. Qing and Z. D. Yang, “A Remark on the Existence of Entire Positive Solutions for a Class of Quasilinear Elliptic Problem,” Oriental Journal of Mathematics, Vol. 3, No. 1, 2010, pp. 43-50.

[8] C. S. Chen, Z. Q. Wang and F. P. Wang,” Existence and Nonexistence of Positive Solutions for Singular p-Laplacian Equation in Rn,” Boundary Value Problems, Vol. 2010, 2010, Article ID 607453, pp. 1-17.

[9] X. G. Zhang and L. S. Liu, “The Existence and Nonexistence of Entire Positive Solutions of Semilinear Elliptic Systems with Gradient Term,” Journal of Mathematical Analysis and Applications, Vol. 371, No. 1, 2010, pp. 300-308. doi:10.1016/j.jmaa.2010.05.029