Existence and Non-Existence Result for Singular Quasilinear Elliptic Equations

ABSTRACT

We prove the existence of a ground state solution for the qusilinear elliptic equation in , under suitable conditions on a locally Holder continuous non-linearity , the non-linearity may exhibit a singularity as . We also prove the non-existence of radially symmetric solutions to the singular elliptic equation in , as where .

We prove the existence of a ground state solution for the qusilinear elliptic equation in , under suitable conditions on a locally Holder continuous non-linearity , the non-linearity may exhibit a singularity as . We also prove the non-existence of radially symmetric solutions to the singular elliptic equation in , as where .

Cite this paper

nullM. Wu and Z. Yang, "Existence and Non-Existence Result for Singular Quasilinear Elliptic Equations,"*Applied Mathematics*, Vol. 1 No. 5, 2010, pp. 351-356. doi: 10.4236/am.2010.15046.

nullM. Wu and Z. Yang, "Existence and Non-Existence Result for Singular Quasilinear Elliptic Equations,"

References

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[2] L. K. Martinson and K. B. Pavlov, “Unsteady Shear Flows of a Conducting Fluid with a Rheological Power Law, Magnit,” Gidrodinamika, Vol. 2, 1971, pp. 50-58.

[3] J. R. Esteban and J. L. Vazquez, “On the Equation of Turbulent Filtration in One-Dimensional Porous Media,” Nonlinear Analysis, Vol. 10, No. 11, 1982, pp. 1303- 1325.

[4] Z. M. Guo, “Existence and Uniqueness of the Positive Radial Solutions for a Class of Quasilinear Elliptic Equations,” Applied Analysis, Vol. 47, No. 3, 1992, pp. 173-190.

[5] Z. M. Guo, “Some Existence and Multiplicity Results for a Class of Quasilinear Elliptic Equatons,” Nonlinear Analysis, Vol. 18, No. 10, 1992, pp. 957-971.

[6] Z. M. Guo and J. R. L. Webb, “Uniqueness of Positive Solutions for Quasilinear Elliptic Equations When a Parameter is Large,” Proceedings of Royal Society of Edinburgh, Edinburgh, Vol. 124, No. 1, 1994, pp. 189- 198.

[7] M. G. Crandall, P. H. Rabinowitz and L. Tartar, “On a Dirichlet Problem with a Singular Nonlinearity,” Commu- nications in Partial Differential Equations, Vol. 2, No. 2, 1977, pp. 193-222.

[8] S. Cui, “Existence and Nonexistence of Positive Solutions for Singular Semilinear Elliptic Boundary Value Problems,” Nonlinear Analysis, Vol. 41, No. 1-2, 2000, pp. 149-176.

[9] A. V. Lair and A. W. Shaker, “Classical and Weak Solutions of a Singular Semilinear Elliptic Problem,” Journal of Mathematical Analysis and Applications, Vol. 211, No. 2, 1997, pp. 371-385.

[10] A. C. Lazer and P. J. McKenna, “On a Singular Nonlinear Elliptic Boundary-Value Problem,” Proceedings of the American Mathematical Society, Vol. 111, No. 3, 1991, pp. 721-730.

[11] H. Maagli and M. Zribi, “Existence and Estimates of Solutions for Singular Nonlinearity Elliptic Problems,” Journal of Mathematical Analysis and Applications, Vol. 263, No. 2, 2001, pp. 522-542.

[12] J. Shi and M. Yao, “Positive Solutions for Elliptic Equations with Singular Nonlinearity,” Electronic Journal of Qualitative Theory of Differential Equations, Vol. 4, 2005, pp. 1-11.

[13] H. Xue and Z. Zhang, “A Remark on Ground State Solutions for Lane-Emden-Fowler Equations with a Convection Term,” Electronic Journal of Differential Equations, Vol. 2007, No. 53, 2007, pp. 1-10.

[14] H. Brezis and S. Kamin, “Sublinear Elliptic Equations in RN,” Manuscripta Mathematica, Vol. 74, No. 1, 1992, pp. 87-106.

[15] S. Wu and H. Yang, “The Existence Theorems for a Class of Sublinear Elliptic Equations in RN,” Acta Mathematica Sinica, Vol. 13, No. 3, 1997, pp. 259-304.

[16] Z. Zhang, “A Remark on the Existence of Entire Solutions of a Singular Semilinear Elliptic Problem,” Journal of Mathematical Analysis and Applications, Vol. 215, No. 2, 1997, pp. 579-582.

[17] Z. Zhang, “A Remark on the Existence of Positive Entire Solutions of a Sublinear Elliptic Problem,” Nonlinear Analysis, Vol. 67, 2007, pp. 147-153.

[18] K. E. Mabrouk, “Entire Bounded Solutions for a Class of Sublinear Elliptic Equations,” Nonlinear Analysis, Vol. 58, No. 1-2, 2004, pp. 205-218.

[19] J. V. Goncalves and C. A. Santos, “Existence and Asymptotic Behavior of Non-Radially Symmetric Ground States of Semilinear Singular Elliptic Equations,” Nonlinear Analysis, Vol. 65, No. 4, 2006, pp. 719-727.

[20] A. Mohammed, “Ground State Solutions for Singular Semilinear Elliptic Equations,” Nonlinear Analysis, Vol. 71, No. 3-4, 2009, pp. 1276-1280.

[1] G. Astrita and G. Marrucci, “Principles of Non-New- tonian Fluid Mechanics,” McGraw-Hill, Rochester, 1974.

[2] L. K. Martinson and K. B. Pavlov, “Unsteady Shear Flows of a Conducting Fluid with a Rheological Power Law, Magnit,” Gidrodinamika, Vol. 2, 1971, pp. 50-58.

[3] J. R. Esteban and J. L. Vazquez, “On the Equation of Turbulent Filtration in One-Dimensional Porous Media,” Nonlinear Analysis, Vol. 10, No. 11, 1982, pp. 1303- 1325.

[4] Z. M. Guo, “Existence and Uniqueness of the Positive Radial Solutions for a Class of Quasilinear Elliptic Equations,” Applied Analysis, Vol. 47, No. 3, 1992, pp. 173-190.

[5] Z. M. Guo, “Some Existence and Multiplicity Results for a Class of Quasilinear Elliptic Equatons,” Nonlinear Analysis, Vol. 18, No. 10, 1992, pp. 957-971.

[6] Z. M. Guo and J. R. L. Webb, “Uniqueness of Positive Solutions for Quasilinear Elliptic Equations When a Parameter is Large,” Proceedings of Royal Society of Edinburgh, Edinburgh, Vol. 124, No. 1, 1994, pp. 189- 198.

[7] M. G. Crandall, P. H. Rabinowitz and L. Tartar, “On a Dirichlet Problem with a Singular Nonlinearity,” Commu- nications in Partial Differential Equations, Vol. 2, No. 2, 1977, pp. 193-222.

[8] S. Cui, “Existence and Nonexistence of Positive Solutions for Singular Semilinear Elliptic Boundary Value Problems,” Nonlinear Analysis, Vol. 41, No. 1-2, 2000, pp. 149-176.

[9] A. V. Lair and A. W. Shaker, “Classical and Weak Solutions of a Singular Semilinear Elliptic Problem,” Journal of Mathematical Analysis and Applications, Vol. 211, No. 2, 1997, pp. 371-385.

[10] A. C. Lazer and P. J. McKenna, “On a Singular Nonlinear Elliptic Boundary-Value Problem,” Proceedings of the American Mathematical Society, Vol. 111, No. 3, 1991, pp. 721-730.

[11] H. Maagli and M. Zribi, “Existence and Estimates of Solutions for Singular Nonlinearity Elliptic Problems,” Journal of Mathematical Analysis and Applications, Vol. 263, No. 2, 2001, pp. 522-542.

[12] J. Shi and M. Yao, “Positive Solutions for Elliptic Equations with Singular Nonlinearity,” Electronic Journal of Qualitative Theory of Differential Equations, Vol. 4, 2005, pp. 1-11.

[13] H. Xue and Z. Zhang, “A Remark on Ground State Solutions for Lane-Emden-Fowler Equations with a Convection Term,” Electronic Journal of Differential Equations, Vol. 2007, No. 53, 2007, pp. 1-10.

[14] H. Brezis and S. Kamin, “Sublinear Elliptic Equations in RN,” Manuscripta Mathematica, Vol. 74, No. 1, 1992, pp. 87-106.

[15] S. Wu and H. Yang, “The Existence Theorems for a Class of Sublinear Elliptic Equations in RN,” Acta Mathematica Sinica, Vol. 13, No. 3, 1997, pp. 259-304.

[16] Z. Zhang, “A Remark on the Existence of Entire Solutions of a Singular Semilinear Elliptic Problem,” Journal of Mathematical Analysis and Applications, Vol. 215, No. 2, 1997, pp. 579-582.

[17] Z. Zhang, “A Remark on the Existence of Positive Entire Solutions of a Sublinear Elliptic Problem,” Nonlinear Analysis, Vol. 67, 2007, pp. 147-153.

[18] K. E. Mabrouk, “Entire Bounded Solutions for a Class of Sublinear Elliptic Equations,” Nonlinear Analysis, Vol. 58, No. 1-2, 2004, pp. 205-218.

[19] J. V. Goncalves and C. A. Santos, “Existence and Asymptotic Behavior of Non-Radially Symmetric Ground States of Semilinear Singular Elliptic Equations,” Nonlinear Analysis, Vol. 65, No. 4, 2006, pp. 719-727.

[20] A. Mohammed, “Ground State Solutions for Singular Semilinear Elliptic Equations,” Nonlinear Analysis, Vol. 71, No. 3-4, 2009, pp. 1276-1280.