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Existence and Non-Existence Result for Singular Quasilinear Elliptic Equations
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Applied Mathematics·Volume 01 (2010)·Pages 351–356·Published 30 November 2010·DOI10.4236/am.2010.15046
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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 .
KeywordsQuasilinear Elliptic EquationsExistenceNon-ExistenceSingular
- G. Astrita and G. Marrucci, “Principles of Non-New- tonian Fluid Mechanics,” McGraw-Hill, Rochester, 1974.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- H. Brezis and S. Kamin, “Sublinear Elliptic Equations in RN,” Manuscripta Mathematica, Vol. 74, No. 1, 1992, pp. 87-106.