## Topological Methods in Nonlinear Analysis

### Unbounded connected component of the positive solutions set of some semi-positone problems

#### Abstract

In this paper, first we obtain some results for structure of positive solutions set of some nonlinear operator equation. Then using these results, we obtain some existence results for positive solutions of the nonlinear operator equation. The method to show our main results is the global bifurcation theory.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 37, Number 2 (2011), 283-302.

Dates
First available in Project Euclid: 20 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461184787

Mathematical Reviews number (MathSciNet)
MR2849824

Zentralblatt MATH identifier
1239.47045

#### Citation

Xian, Xu; Jingxian, Sun. Unbounded connected component of the positive solutions set of some semi-positone problems. Topol. Methods Nonlinear Anal. 37 (2011), no. 2, 283--302. https://projecteuclid.org/euclid.tmna/1461184787

#### References

• R. P. Agarwal and D. O' Regan, Nonpositone discrete boundary value problems , Nonlinear Anal. (2000, 39), 207–215 \ref\key 2
• A. Ambrosetti, D. Arcoya and B. Buffoni, Positive solutions for some semipositone problems via bifurcation theory , Differential Integral Equations, 7 (1994), 655–663 \ref\key 3
• V. Anuradha, D. D. Hai and R. Shivaji, Existence results for superlinear semipositone BVP's , Proc. Amer. Math. Soc. (1996, 124), 757–763 \ref\key 4
• D. Arcoya and A. Zertiti, Existence and nonexistence of radially symmetric nonnegative solutions for a class of semipositone problems in an annulus , Rend. Math. (8), 14 (1994), 625–646 \ref\key 5
• A. Castro, S. Gadam and R. Shivaji, Branches of radial solutions for semipositone problems , J. Differential Equations, 120 (1995), 30–45 \ref\key 6 ––––, Positive solution curves of semipositone problems with concave nonlinearities , Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 921–934 \ref\key 7
• A. Castro M. Hassanpour and R. Shivaji, Uniqueness of nonnegative solutions for a semipositone problem with concave nonlinearity , Comm. Partial Differential Equations, 20 (1995), 1927–1936 \ref\key 8
• A. Castro, C. Maya and R. Shivaji, Nonlinear eigenvalue problems with semipositone , Electron. J. Differential Equations Conf. (2000, 5 ), 33–49 \ref\key 9
• A. Castro and R. Shivaji, Nonnegative solutions for a class of nonpositone problems , Proc. Roy. Soc. Edinburgh (1988, 108 (A) ), 291–302 \ref\key 10 ––––, Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric , Comm. Partial Differential Equations, 14 (1989), 1091–1100 \ref\key 11 ––––, Nonnegative solutions for a class of radially symmetric nonpositone problems , Proc. Amer. Math. Soc., 106 (1989) \ref\key 12 ––––, Positive solutions for a concave semipositone dirichlet problem , Nonlinear Anal., 31 (1998), 91–98 \ref\key 13 ––––, Semipositone Problems, Semigroups of Linear and Nonlinear Operators and Applications , Kluwer Academic Publishers, New York (J. A. Goldstein and G. Goldstein, eds.)(1993), 109–119. (invited review paper) \ref\key 14
• E. N. Dancer, Global solution branches for positive mappings , Arch. Rational Mech. Anal., 52 (1973), 181–192 \ref\key 15
• K. Deimling, Nonlinear Functional Analysis, Springer–Verlag, Berlin–Heidelberg–New York (1985) \ref\key 16
• M. A. Krasnosel'skiĭ and P. P. Zabreĭko, Geometrical Methods of Nonlinear Analysis, Springer–Verlag, Berlin–Heidelberg–New York–Tokyo (1984) \ref\key 17
• P. L. Lions, On the existence of positive solutions of semilinear elliptic equations , SIAM Rev., 24 (1982), 441–467 \ref\key 18
• J. Sun, The existence of positive solutions for nonlinear Hammerstein integral equations and their applications , (Chinese, Chinese Ann. Math. Ser. A, 9 (1988), 90–96) \ref\key 19
• J. Sun, X. Xu and D. O'Regan, Nodal solutions for $m$-point boundary value problems using bifurcation methods , Nonlinear Anal., 68 (2008), 3034–3046 \ref\key 20
• S. Unsurangsie, Existence of a solution for a wave equation and elliptic Dirichlet problem , Ph.D. Thesis (1988, University of North Texas) \ref\key 21
• X. Xu and D. O'Regan, Existence of positive solutions for operator equations and spplications to semipositone problems , Positivity (2006, 10 ), 315–328