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  • Research Article
  • Open Access

Existence and Asymptotic Behavior of Positive Solutions to -Laplacian Equations with Singular Nonlinearities

Journal of Inequalities and Applications20072007:019349

  • Received: 17 July 2007
  • Accepted: 27 August 2007
  • Published:


This paper investigates the -Laplacian equations with singular nonlinearities in , on , where is called -Laplacian. The existence of positive solutions is given, and the asymptotic behavior of solutions near boundary is discussed.


  • Asymptotic Behavior
  • Laplacian Equation
  • Singular Nonlinearity


Authors’ Affiliations

Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan, 450002, China


  1. Chen Y, Levine S, Rao M: Variable exponent, linear growth functionals in image restoration. SIAM Journal on Applied Mathematics 2006,66(4):1383–1406.MathSciNetView ArticleGoogle Scholar
  2. Rúzicka M: Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics. Volume 1748. Springer, Berlin, Germany; 2000:xvi+176.Google Scholar
  3. Fan X-L, Zhao D: On the spacesand. Journal of Mathematical Analysis and Applications 2001,263(2):424–446. 10.1006/jmaa.2000.7617MathSciNetView ArticleMATHGoogle Scholar
  4. Fan X-L, Zhao D: The quasi-minimizer of integral functionals withgrowth conditions. Nonlinear Analysis: Theory, Methods & Applications 2000,39(7):807–816. 10.1016/S0362-546X(98)00239-9MathSciNetView ArticleMATHGoogle Scholar
  5. Fan X-L, Zhang Q-H: Existence of solutions for-Laplacian Dirichlet problem. Nonlinear Analysis: Theory, Methods & Applications 2003,52(8):1843–1852. 10.1016/S0362-546X(02)00150-5MathSciNetView ArticleMATHGoogle Scholar
  6. Fan X-L, Zhang Q, Zhao D: Eigenvalues of-Laplacian Dirichlet problem. Journal of Mathematical Analysis and Applications 2005,302(2):306–317. 10.1016/j.jmaa.2003.11.020MathSciNetView ArticleMATHGoogle Scholar
  7. Fan X-L: Globalregularity for variable exponent elliptic equations in divergence form. Journal of Differential Equations 2007,235(2):397–417. 10.1016/j.jde.2007.01.008MathSciNetView ArticleMATHGoogle Scholar
  8. Kováčik O, Rákosník J: On spacesand. Czechoslovak Mathematical Journal 1991,41(4):592–618.MathSciNetGoogle Scholar
  9. Marcellini P: Regularity and existence of solutions of elliptic equations with-growth conditions. Journal of Differential Equations 1991,90(1):1–30. 10.1016/0022-0396(91)90158-6MathSciNetView ArticleMATHGoogle Scholar
  10. Zhang Q: A strong maximum principle for differential equations with nonstandard-growth conditions. Journal of Mathematical Analysis and Applications 2005,312(1):24–32. 10.1016/j.jmaa.2005.03.013MathSciNetView ArticleMATHGoogle Scholar
  11. Zhang Q: Existence of positive solutions for a class of-Laplacian systems. Journal of Mathematical Analysis and Applications 2007,333(2):591–603. 10.1016/j.jmaa.2006.11.037MathSciNetView ArticleMATHGoogle Scholar
  12. Zhang Q: Existence of positive solutions for elliptic systems with nonstandard-growth conditions via sub-supersolution method. Nonlinear Analysis: Theory, Methods & Applications 2007,67(4):1055–1067. 10.1016/ ArticleMATHGoogle Scholar
  13. Zhang Q: Oscillatory property of solutions for-laplacian equations. Journal of Inequalities and Applications 2007, 2007: 8 pages.Google Scholar
  14. Baxley JV: Some singular nonlinear boundary value problems. SIAM Journal on Mathematical Analysis 1991,22(2):463–479. 10.1137/0522030MathSciNetView ArticleMATHGoogle Scholar
  15. Cui S: Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2000,41(1–2):149–176. 10.1016/S0362-546X(98)00271-5View ArticleMATHGoogle Scholar
  16. Fink AM, Gatica JA, Hernández GE, Waltman P: Approximation of solutions of singular second order boundary value problems. SIAM Journal on Mathematical Analysis 1991,22(2):440–462. 10.1137/0522029MathSciNetView ArticleMATHGoogle Scholar
  17. Lü H, Bai Z: Positive radial solutions of a singular elliptic equation with sign changing nonlinearities. Applied Mathematics Letters 2006,19(6):555–567. 10.1016/j.aml.2005.08.002MathSciNetView ArticleMATHGoogle Scholar
  18. Shi J, Yao M: On a singular nonlinear semilinear elliptic problem. Proceedings of the Royal Society of Edinburgh. Section A 1998,128(6):1389–1401. 10.1017/S0308210500027384MathSciNetView ArticleMATHGoogle Scholar