Entire Large Solutions of Quasilinear Elliptic Equations of Mixed Type

ABSTRACT

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.

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.

Cite this paper

nullH. Qin and Z. Yang, "Entire Large Solutions of Quasilinear Elliptic Equations of Mixed Type,"*Applied Mathematics*, Vol. 1 No. 4, 2010, pp. 293-300. doi: 10.4236/am.2010.14038.

nullH. Qin and Z. Yang, "Entire Large Solutions of Quasilinear Elliptic Equations of Mixed Type,"

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,” 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.

[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.