- Shobha Oruganti
^{1}and - R Shivaji
^{2}Email author

**2006**:87483

**DOI: **10.1155/BVP/2006/87483

© S. Oruganti and R. Shivaji 2006

**Received: **22 September 2005

**Accepted: **10 November 2005

**Published: **23 February 2006

## 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.

## Authors’ Affiliations

## References

- 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 GiorgiMathSciNetMATHGoogle Scholar - 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/S030821050003167XMathSciNetView ArticleMATHGoogle Scholar - 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 - Chhetri M, Oruganti S, Shivaji R:
**Positive solutions for classes of-Laplacian equations.***Differential and Integral Equations*2003,**16**(6):757–768.MathSciNetMATHGoogle Scholar - 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)90006-8MathSciNetView ArticleMATHGoogle Scholar - 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.MathSciNetMATHGoogle Scholar - 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)00258-8MathSciNetView ArticleMATHGoogle Scholar - 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 - 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.1994.43.43053MathSciNetView ArticleMATHGoogle Scholar - Hai DD:
**On a class of sublinear quasilinear elliptic problems.***Proceedings of the American Mathematical Society*2003,**131**(8):2409–2414. 10.1090/S0002-9939-03-06874-6MathSciNetView ArticleMATHGoogle Scholar - 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)00028-7MathSciNetView ArticleMATHGoogle Scholar - 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/S0002-9947-02-03005-2MathSciNetView ArticleMATHGoogle Scholar - 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/S1085337504311097MathSciNetView ArticleMATHGoogle Scholar

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