Entire Large Solutions of Quasilinear Elliptic Equations of Mixed Type ()

Hongxia Qin, Zuodong Yang

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**DOI: **10.4236/am.2010.14038
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In this paper, the existence and nonexistence of nonnegative entire large solutions for the quasilinear elliptic equation are established, where , and are nondecreasing and vanish at the origin. The locally H lder continuous functions and are nonnegative. We extend results previously obtained for special cases of and g.

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Qin, H. and Yang, Z. (2010) Entire Large Solutions of Quasilinear Elliptic Equations of Mixed Type. *Applied Mathematics*, **1**, 293-300. doi: 10.4236/am.2010.14038.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | G. Astrita and G. Marruci, “Principles of Non-Newtonian Fluid Mechanics,” McGraw-Hill, 1974. |

[2] | L. k. Martinson and K. B. Pavlov, “Unsteady Shear Flows of a Conducting Fluid with a Rheological Power Law,” Magnitnaya Gidrodinamika, Vol. 7, No. 2, 1971, pp.50 -58. |

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

[4] | A. S. Kalashnikov, “On a Nonlinear Equation Appearing in the Theory of Nonstationary Filtration,” Trudy Seminara I.G. Petrovski, Russian, 1978. |

[5] | S. L. Phhozaev, “The Dirichlet Problem for the Equation ,” Soviet mathematics-Doklady, Vol. 1, No. 2, 1960, pp. 1143-1146. |

[6] | A. C. Lazer and P. J. Mckenna, “On a Problem of Bieberbach and Rademacher,” Non-Linear Analysis archive, Vol. 21, No. 5, 1993, pp. 327-335. |

[7] | K.-S. Cheng and W.-M. Ni, “On the Structure of the Conformal Scalar Curvature Equation on RN,” Indiana University Mathematic Journal, Vol. 41, No. 1, 1992, pp. 261-278. |

[8] | V. Anuradha, C. Brown and R. Shivaji, “Explosive Nonnegative Solutions to Two Point Boundary Value Problems,” Non-Linear Analysis archive, Vol. 26, No. 3, 1996, pp. 613-630. |

[9] | S.-H. Wang, “Existence and Multiplicity of Boundary Blow-Up Nonnegative Solutions to Two-Point Boundary Value Problems,” Non-Linear Analysis archive, Vol. 42, No. 1, 2000, pp. 139-162. |

[10] | G. Diaz and R. Letelier, “Explosive Solutions of Quasilinear Elliptic Equations: Existence and Uniqueness,” Non-Linear Analysis archive, Vol. 20, No. 1, 1993, pp. 97 -125. |

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

[12] | A. C. Lazer and P. J. McKenna, “On Singular Boundary Value Problems for the Monge-Ampere Operator,” Journal of Mathematical Analysis Applications, Vol. 197, No. 2, 1996, pp. 341-362. |

[13] | L. Bieberbach, “ und die automorphen Funktionen,” Mathematische Annalen, Vol. 77, No. 1, 1916, pp. 173 -212. |

[14] | M. Marcus and L. Veron, “Uniqueness of Solutions with Blow-Up at the Boundary for a Class of Nonlinear Elliptic Equation,” Comptes rendus de l'Académie des sciences, Vol. 317, No. 2, 1993, pp. 559-563. |

[15] | S. L. Pohozaev, “The Dirichlet Problem for the Equation ,” Soviet mathematics-Doklady, Vol. 1, No. 2, 1960, pp. 1143-1146. |

[16] | M. R. Posteraro, “On the Solutions of the Equation Blowing up on the Boundary,” Comptes rendus de l'Académie des sciences, Vol. 322, No. 2, 1996, pp. 445-450. |

[17] | H. Rademacher, “Einige Besondere Problem Partieller Differentialgleichungen,” In: Die Differential-und Integralgleichungen, der Mechanik und Physikl, Rosenberg, New York, 1943, pp. 838-845. |

[18] | J. B. Keller, “On Solutions of ,” Communications on Pure and Applied Mathematics, Vol. 10, No. 4, 1957, pp. 503-510. |

[19] | V. A. Kondrat'ev and V. A. Nikishken, “Asymptotics, near the Boundary, of a Singular Boundary-Value Problem for a Semilinear Elliptic Equation,” Differential Equations, Vol. 26, No. 1, 1990, pp. 345-348. |

[20] | C. Loewner and L. Nirenberg, “Partial Differential Equations Invariant under Conformal or Projective Transformations,” In: Contributions to Analysis (A Collection of Paper Dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 245-272. |

[21] | E. B. Dynkin, “Superprocesses and Partial Differential Equations,” Annals of Probability, Vol. 21, No. 3, 1993, pp. 1185-1262. |

[22] | E. B. Dynkin and S. E. Kuznetsov, “Superdiffusions and Removable Singularities for Quasilinear Partial Differential Equations,” Communications on Pure and Applied Mathematics, Vol. 49, No. 2, 1996, pp. 125-176. |

[23] | A. V. Lair, “Large Solutions of Mixed Sublinear/Superlinear Elliptic Equations,” Journal of Mathematical Analysis Applications, Vol. 346, No. 1, 2008, pp. 99-106. |

[24] | A. V. Lair and A. Mohammed, “Entire Large Solutions of Semilinear Elliptic Equations of Mixed Type,” Communications on Pure and Applied Analysis, Vol. 8, No. 5, 2009, pp. 1607-1618. |

[25] | Q. S. Lu, Z. D. Yang and E. H. Twizell, “Existence of Entire Explosive Positive Solutions of Quasi-linear Elliptic Equations,” Applied Mathematics and Computation, Vol. 148, No. 2, 2004, pp. 359-372. |

[26] | Z. D. Yang, “Existence of Explosive Positive Solutions of Quasilinear Elliptic Equations,” Applied Mathematics and Computation, Vol. 177, No. 2, 2006, pp. 581-588. |

[27] | J. L. Yuan and Z. D. Yang, “Existence of Large Solutions for a Class of Quasilinear Elliptic Equations,” Applied Mathematics and Computation, Vol. 201, No. 2, 2008, pp. 852-858. |

[28] | A. V. Lair, “Large Solutions of Semilinear Elliptic Equations under the Keller-Osserman Condition,” Journal of Mathematical Analysis Applications, Vol. 328, No. 2, 2007, pp. 1247-1254. |

[29] | Z. D. Yang, B. Xu and M. Z. Wu, “Existence of Positive Boundary Blow-up Solutions for Quasilinear Elliptic Equations via Sub and Supersolutions,” Applied Mathematics and Computation, Vol. 188, No. 1, 2007, pp. 492- 498. |

[30] | Z. D. Yang, “Existence of Entire Explosive Positive Radial Solutions for a Class of Quasilinear Elliptic Systems,” Journal of Mathematical Analysis Applications, Vol. 288, No. 2, 2003, pp. 768-783. |

[31] | H. H. Yin and Z. D. Yang, “New Results on the Existence of Bounded Positive Entire Solutions for Quasilinear Elliptic Systems,” Applied Mathematics and Computation, Vol. 190, No. 1, 2007, pp. 441-448. |

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

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