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Existence results for classes of -Laplacian semipositone equations

Boundary Value Problems volume 2006, Article number: 87483 (2006) Cite this article

Abstract

We study positive solutions to classes of boundary value problems of the form in on, where denotes the-Laplacian operator defined by;, is a parameter, is a bounded domain in; with of class and connected (if, we assume that is a bounded open interval), and for some (semipositone problems). In particular, we first study the case when where is a parameter and is a function such that, for and for. We establish positive constants and such that the above equation has a positive solution when and. Next we study the case when (logistic equation with constant yield harvesting) where and is a function that is allowed to be negative near the boundary of. Here is a function satisfying for,, and. We establish a positive constant such that the above equation has a positive solution when Our proofs are based on subsuper solution techniques.

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References

  1. Berestycki H, Caffarelli LA, Nirenberg L: Further qualitative properties for elliptic equations in unbounded domains. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1997,25(1–2):69–94. dedicated to E. De Giorgi

    MathSciNet  MATH  Google Scholar 

  2. Brown KJ, Shivaji R: Simple proofs of some results in perturbed bifurcation theory. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1982,93(1–2):71–82. 10.1017/S030821050003167X

    Article  MathSciNet  MATH  Google Scholar 

  3. Castro A, Maya C, Shivaji R: Nonlinear eigenvalue problems with semipositone structure. In Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, Fla, 1999), 2000, Texas, Electron. J. Differ. Equ. Conf.. Volume 5. Southwest Texas State University; 33–49.

    Google Scholar 

  4. Chhetri M, Oruganti S, Shivaji R: Positive solutions for classes of-Laplacian equations. Differential and Integral Equations 2003,16(6):757–768.

    MathSciNet  MATH  Google Scholar 

  5. Clément Ph, Peletier LA: An anti-maximum principle for second-order elliptic operators. Journal of Differential Equations 1979,34(2):218–229. 10.1016/0022-0396(79)90006-8

    Article  MathSciNet  MATH  Google Scholar 

  6. Clément Ph, Sweers G: Existence and multiplicity results for a semilinear elliptic eigenvalue problem. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1987,14(1):97–121.

    MathSciNet  MATH  Google Scholar 

  7. Drábek P, Hernández J: Existence and uniqueness of positive solutions for some quasilinear elliptic problems. Nonlinear Analysis 2001,44(2):189–204. 10.1016/S0362-546X(99)00258-8

    Article  MathSciNet  MATH  Google Scholar 

  8. Drábek P, Krejčí P, Takáč P: Nonlinear Differential Equations, Chapman & Hall/CRC Research Notes in Mathematics. Volume 404. Chapman & Hall/CRC, Florida; 1999:vi+196.

    Google Scholar 

  9. Fleckinger-Pellé J, Takáč P: Uniqueness of positive solutions for nonlinear cooperative systems with the-Laplacian. Indiana University Mathematics Journal 1994,43(4):1227–1253. 10.1512/iumj.1994.43.43053

    Article  MathSciNet  MATH  Google Scholar 

  10. Hai DD: On a class of sublinear quasilinear elliptic problems. Proceedings of the American Mathematical Society 2003,131(8):2409–2414. 10.1090/S0002-9939-03-06874-6

    Article  MathSciNet  MATH  Google Scholar 

  11. Hai DD, Shivaji R: Existence and uniqueness for a class of quasilinear elliptic boundary value problems. Journal of Differential Equations 2003,193(2):500–510. 10.1016/S0022-0396(03)00028-7

    Article  MathSciNet  MATH  Google Scholar 

  12. Oruganti S, Shi J, Shivaji R: Diffusive logistic equation with constant yield harvesting. I. Steady states. Transactions of the American Mathematical Society 2002,354(9):3601–3619. 10.1090/S0002-9947-02-03005-2

    Article  MathSciNet  MATH  Google Scholar 

  13. Oruganti S, Shi J, Shivaji R: Logistic equation wtih the -Laplacian and constant yield harvesting. Abstract and Applied Analysis 2004,2004(9):723–727. 10.1155/S1085337504311097

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, School of Science, The Behrend College, Penn State Erie, Erie, PA, 16563, USA

    Shobha Oruganti

  2. Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS, 39762, USA

    R Shivaji

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  1. Shobha Oruganti
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  2. R Shivaji
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Correspondence to R Shivaji.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Oruganti, S., Shivaji, R. Existence results for classes of -Laplacian semipositone equations. Bound Value Probl 2006, 87483 (2006). https://doi.org/10.1155/BVP/2006/87483

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