Liouville-Type Theorems for Some Integral Systems

ABSTRACT

In this paper, Liouville-type theorems of nonnegative solutions for some elliptic integral systems are considered. We use a new type of moving plane method introduced by Chen-Li-Ou. Our new ingredient is the use of Stein-Weiss inequality instead of Maximum Principle.

In this paper, Liouville-type theorems of nonnegative solutions for some elliptic integral systems are considered. We use a new type of moving plane method introduced by Chen-Li-Ou. Our new ingredient is the use of Stein-Weiss inequality instead of Maximum Principle.

Cite this paper

nullZ. Zhang, "Liouville-Type Theorems for Some Integral Systems,"*Applied Mathematics*, Vol. 1 No. 2, 2010, pp. 94-100. doi: 10.4236/am.2010.12012.

nullZ. Zhang, "Liouville-Type Theorems for Some Integral Systems,"

References

[1] W. X. Chen and C. M. Li, “Classification of Positive Solutions for Nonlinear Differential and Integral Systems with Critical Exponents,” Acta Mathematica Scientia, in press.

[2] W. X. Chen, C. M. Li and B. Ou, “Classification of Solutions for an Integral Equation,” Communications on Pure and Applied Mathematics, Vol. 59, No. 3, 2006, pp. 330- 343.

[3] L. Caffarelli, B. Gidas and J. Spruck, “Asymptotic Symmetry and Local Behavior of Semilinear Elliptic Equations with Critical Sobolev Growth,” Communications on Pure and Applied Mathematics, Vol. 42, No. 3, 1989, pp. 271-297.

[4] B. Gidas and J. Spruck, “Global and Local Behaviour of Positive Solutions of Nonlinear Elliptic Equations,” Communications on Pure and Applied Mathematics, Vol. 34, No. 6, 1981, pp. 525-598.

[5] S. H. Chen and G. Z. Lu, “Existence and Nonexistence of Positive Radial Solutions for a Class of Semilinear Elliptic System,” Nonlinear Analysis, Vol. 38, No. 7, 1999, pp. 919-932.

[6] E. Mitidieri, “Nonexistence of Positive Solutions of Semilinear Elliptic Systems in ,” Differential and Integral Equations, Vol. 9, No. 3, 1996, pp. 465-479.

[7] E. Lieb, “Sharp Constants in the Hardy-Littlewood- Sobolev and Related Inequalities,” Annals of Mathematics, Vol. 118, 1983, pp. 349-374.

[8] D. G. de Figueiredo and P. L. Felmer, “A Liouville-Type Theorem for Elliptic Systems,” Annali della Scuola Normale Superiore di Pisa XXI, Vol. 21, No. 3, 1994, pp. 387-397.

[9] J. Busca and R. Manásevich, “A Liouville-Type Theorem for Lane-Emden Systems,” Indiana University Mathematics Journal, Vol. 51, No. 1, 2002, pp. 37-51.

[10] Z. C. Zhang, W. M. Wang and K. T. Li, “Liouville-Type Theorems for Semilinear Elliptic Systems,” Journal of Partial Differential Equations, Vol. 18, No. 4, 2005, pp. 304-310.

[11] Y. Y. Li and M. Zhu, “Uniqueness Theorems through the Method of Moving Spheres,” Duke Mathematical Journal, Vol. 80, No. 2, 1995, pp. 383-417.

[12] Y. Y. Li and L. Zhang, “Liouville-Type Theorems and Harnack-Type Inequalities for Semilinear Elliptic Equations,” Journal d’Analyse Mathematique, Vol. 90, 2003, pp. 27-87.

[13] W. X. Chen, C. M. Li and B. Ou, “Classification of Solutions for a System of Integral Equations,” Communications in Partial Differential Equations, Vol. 30, No. 1-2, 2005, pp. 59-65.

[14] W. X. Chen, C. M. Li and B. Ou, “Qualitative Properties of Solutions for an Integral Equation,” Discrete and Continuous Dynamical Systems, Vol. 12, No. 2, 2005, pp. 347-354.

[15] L. Ma and D. Z. Chen, “A Liouville Type Theorem for an Integral System,” Communications on Pure and Applied Analysis, Vol. 5, No. 4, 2006, pp. 855-859.

[16] C. S. Lin, “A Classification of Solutions of a Conformally Invariant Fourth Order Equation in ,” Commentarii Mathematici Helvetici, Vol. 73, No. 2, 1998, pp. 206-231.

[17] L. A. Peletier and R. C. A. M. van der Vorst, “Existence and Non-Existence of Positive Solutions of Non-Linear Elliptic Systems and the Biharmonic Equation,” Differential and Integral Equations, Vol. 5, No. 4, 1992, pp. 747- 767.

[18] J. Serrin and H. Zou, “The Existence of Positive Entire Solutions of Elliptic Hamitonian Systems,” Communications in Partial Differential Equations, Vol. 23, No. 3-4, 1998, pp. 577-599.

[19] M. A. Souto, “A Priori Estimates and Existence of Positive Solutions of Semilinear Elliptic Systems,” Differential and Integral Equations, Vol. 8, No. 5, 1995, pp. 1245- 1258.

[20] E. M. Stein, “Singular Integral and Differentiability Properties of Functions,” Princeton University Press, Princeton, 1970.

[21] Z. C. Zhang and Z. M. Guo, “Structure of Nontrivial Nonnegative Solutions of Singularly Perturbed Quasilinear Dirichlet Problems,” Mathematische Nachrichten, Vol. 280, No. 13-14, 2007, pp. 1620-1639.

[22] Z. C. Zhang and K. T. Li, “Spike-Layered Solutions of Singularly Perturbed Quasilinear Dirichlet Problems,” Journal of Mathematical Analysis and Applications, Vol. 283, No. 2, 2003, pp. 667-680.

[23] Z. C. Zhang and K. T. Li, “Spike-Layered Solutions with Compact Support to Some Singularly Perturbed Quasilinear Elliptic Problems in General Smooth Domains,” Journal of Computational and Applied Mathematics, Vol. 162, No. 2, 2004, pp. 327-340.

[24] L. P. Zhu, Z. C. Zhang and W. M. Wang, “On the Positive Solutions for a Class of Semilinear Elliptic Systems,” Mathematical Applications, Vol. 19, No. 2, 2006, pp. 440- 445.

[1] W. X. Chen and C. M. Li, “Classification of Positive Solutions for Nonlinear Differential and Integral Systems with Critical Exponents,” Acta Mathematica Scientia, in press.

[2] W. X. Chen, C. M. Li and B. Ou, “Classification of Solutions for an Integral Equation,” Communications on Pure and Applied Mathematics, Vol. 59, No. 3, 2006, pp. 330- 343.

[3] L. Caffarelli, B. Gidas and J. Spruck, “Asymptotic Symmetry and Local Behavior of Semilinear Elliptic Equations with Critical Sobolev Growth,” Communications on Pure and Applied Mathematics, Vol. 42, No. 3, 1989, pp. 271-297.

[4] B. Gidas and J. Spruck, “Global and Local Behaviour of Positive Solutions of Nonlinear Elliptic Equations,” Communications on Pure and Applied Mathematics, Vol. 34, No. 6, 1981, pp. 525-598.

[5] S. H. Chen and G. Z. Lu, “Existence and Nonexistence of Positive Radial Solutions for a Class of Semilinear Elliptic System,” Nonlinear Analysis, Vol. 38, No. 7, 1999, pp. 919-932.

[6] E. Mitidieri, “Nonexistence of Positive Solutions of Semilinear Elliptic Systems in ,” Differential and Integral Equations, Vol. 9, No. 3, 1996, pp. 465-479.

[7] E. Lieb, “Sharp Constants in the Hardy-Littlewood- Sobolev and Related Inequalities,” Annals of Mathematics, Vol. 118, 1983, pp. 349-374.

[8] D. G. de Figueiredo and P. L. Felmer, “A Liouville-Type Theorem for Elliptic Systems,” Annali della Scuola Normale Superiore di Pisa XXI, Vol. 21, No. 3, 1994, pp. 387-397.

[9] J. Busca and R. Manásevich, “A Liouville-Type Theorem for Lane-Emden Systems,” Indiana University Mathematics Journal, Vol. 51, No. 1, 2002, pp. 37-51.

[10] Z. C. Zhang, W. M. Wang and K. T. Li, “Liouville-Type Theorems for Semilinear Elliptic Systems,” Journal of Partial Differential Equations, Vol. 18, No. 4, 2005, pp. 304-310.

[11] Y. Y. Li and M. Zhu, “Uniqueness Theorems through the Method of Moving Spheres,” Duke Mathematical Journal, Vol. 80, No. 2, 1995, pp. 383-417.

[12] Y. Y. Li and L. Zhang, “Liouville-Type Theorems and Harnack-Type Inequalities for Semilinear Elliptic Equations,” Journal d’Analyse Mathematique, Vol. 90, 2003, pp. 27-87.

[13] W. X. Chen, C. M. Li and B. Ou, “Classification of Solutions for a System of Integral Equations,” Communications in Partial Differential Equations, Vol. 30, No. 1-2, 2005, pp. 59-65.

[14] W. X. Chen, C. M. Li and B. Ou, “Qualitative Properties of Solutions for an Integral Equation,” Discrete and Continuous Dynamical Systems, Vol. 12, No. 2, 2005, pp. 347-354.

[15] L. Ma and D. Z. Chen, “A Liouville Type Theorem for an Integral System,” Communications on Pure and Applied Analysis, Vol. 5, No. 4, 2006, pp. 855-859.

[16] C. S. Lin, “A Classification of Solutions of a Conformally Invariant Fourth Order Equation in ,” Commentarii Mathematici Helvetici, Vol. 73, No. 2, 1998, pp. 206-231.

[17] L. A. Peletier and R. C. A. M. van der Vorst, “Existence and Non-Existence of Positive Solutions of Non-Linear Elliptic Systems and the Biharmonic Equation,” Differential and Integral Equations, Vol. 5, No. 4, 1992, pp. 747- 767.

[18] J. Serrin and H. Zou, “The Existence of Positive Entire Solutions of Elliptic Hamitonian Systems,” Communications in Partial Differential Equations, Vol. 23, No. 3-4, 1998, pp. 577-599.

[19] M. A. Souto, “A Priori Estimates and Existence of Positive Solutions of Semilinear Elliptic Systems,” Differential and Integral Equations, Vol. 8, No. 5, 1995, pp. 1245- 1258.

[20] E. M. Stein, “Singular Integral and Differentiability Properties of Functions,” Princeton University Press, Princeton, 1970.

[21] Z. C. Zhang and Z. M. Guo, “Structure of Nontrivial Nonnegative Solutions of Singularly Perturbed Quasilinear Dirichlet Problems,” Mathematische Nachrichten, Vol. 280, No. 13-14, 2007, pp. 1620-1639.

[22] Z. C. Zhang and K. T. Li, “Spike-Layered Solutions of Singularly Perturbed Quasilinear Dirichlet Problems,” Journal of Mathematical Analysis and Applications, Vol. 283, No. 2, 2003, pp. 667-680.

[23] Z. C. Zhang and K. T. Li, “Spike-Layered Solutions with Compact Support to Some Singularly Perturbed Quasilinear Elliptic Problems in General Smooth Domains,” Journal of Computational and Applied Mathematics, Vol. 162, No. 2, 2004, pp. 327-340.

[24] L. P. Zhu, Z. C. Zhang and W. M. Wang, “On the Positive Solutions for a Class of Semilinear Elliptic Systems,” Mathematical Applications, Vol. 19, No. 2, 2006, pp. 440- 445.