# On the uniqueness of positive solutions for two-point boundary value problems of Emden-Fowler differential equations

Mathematica Bohemica (2010)

- Volume: 135, Issue: 2, page 189-198
- ISSN: 0862-7959

## Access Full Article

top## Abstract

top## How to cite

topTanaka, Satoshi. "On the uniqueness of positive solutions for two-point boundary value problems of Emden-Fowler differential equations." Mathematica Bohemica 135.2 (2010): 189-198. <http://eudml.org/doc/38123>.

@article{Tanaka2010,

abstract = {The two-point boundary value problem \[ u^\{\prime \prime \} + h(x) u^p = 0, \quad a < x < b, \qquad u(a) = u(b) = 0 \]
is considered, where $p>1$, $h \in C^1[0,1]$ and $h(x)>0$ for $a \le x \le b$. The existence of positive solutions is well-known. Several sufficient conditions have been obtained for the uniqueness of positive solutions. On the other hand, a non-uniqueness example was given by Moore and Nehari in 1959. In this paper, new uniqueness results are presented.},

author = {Tanaka, Satoshi},

journal = {Mathematica Bohemica},

keywords = {uniqueness; positive solution; two-point boundary value problem; Emden-Fowler equation; uniqueness; positive solution; two-point boundary value problem; Emden-Fowler equation},

language = {eng},

number = {2},

pages = {189-198},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {On the uniqueness of positive solutions for two-point boundary value problems of Emden-Fowler differential equations},

url = {http://eudml.org/doc/38123},

volume = {135},

year = {2010},

}

TY - JOUR

AU - Tanaka, Satoshi

TI - On the uniqueness of positive solutions for two-point boundary value problems of Emden-Fowler differential equations

JO - Mathematica Bohemica

PY - 2010

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 135

IS - 2

SP - 189

EP - 198

AB - The two-point boundary value problem \[ u^{\prime \prime } + h(x) u^p = 0, \quad a < x < b, \qquad u(a) = u(b) = 0 \]
is considered, where $p>1$, $h \in C^1[0,1]$ and $h(x)>0$ for $a \le x \le b$. The existence of positive solutions is well-known. Several sufficient conditions have been obtained for the uniqueness of positive solutions. On the other hand, a non-uniqueness example was given by Moore and Nehari in 1959. In this paper, new uniqueness results are presented.

LA - eng

KW - uniqueness; positive solution; two-point boundary value problem; Emden-Fowler equation; uniqueness; positive solution; two-point boundary value problem; Emden-Fowler equation

UR - http://eudml.org/doc/38123

ER -

## References

top- Coffman, C. V., 10.1016/0022-0396(67)90009-5, J. Differ. Equations 3 (1967), 92-111. (1967) MR0204755DOI10.1016/0022-0396(67)90009-5
- Dalmasso, R., 10.1090/S0002-9939-1995-1301018-5, Proc. Amer. Math. Soc. 123 (1995), 3417-3424 . (1995) Zbl0857.34034MR1301018DOI10.1090/S0002-9939-1995-1301018-5
- Korman, P., On the multiplicity of solutions of semilinear equations, Math. Nachr. 229 (2001), 119-127 . (2001) Zbl0999.35028MR1855158
- Kwong, M. K., On the Kolodner-Coffman method for the uniqueness problem of Emden-Fowler BVP, Z. Angew. Math. Phys. 41 (1990), 79-104 . (1990) Zbl0708.34026MR1036511
- Kwong, M. K., Uniqueness results for Emden-Fowler boundary value problems, Nonlinear Anal. 16 (1991) 435-454 . MR1093379
- Moore, R., Nehari, Z., Nonoscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc. 93 (1959), 30-52 . (1959) MR0111897
- Moroney, R. M., Note on a theorem of Nehari, Proc. Amer. Math. Soc. 13 (1962) 407-410 . Zbl0115.30601MR0148983
- Naito, M., Naito, Y., Solutions with prescribed numbers of zeros for nonlinear second order differential equations, Funkcial. Ekvac. 37 (1994) 505-520 . Zbl0820.34019MR1311557
- Naito, Y., Uniqueness of positive solutions of quasilinear differential equations, Differ. Int. Equations 8 (1995), 1813-1822 . (1995) Zbl0831.34028MR1347982
- Ni, W.-M., Nussbaum, R. D., Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u,r)=0$, Comm. Pure Appl. Math. 38 (1985), 67-108 . (1985) Zbl0581.35021MR0768105
- Tanaka, S., On the uniqueness of solutions with prescribed numbers of zeros for a two-point boundary value problem, Differ. Int. Equations 20 (2007), 93-104. (2007) Zbl1212.34040MR2282828
- Tanaka, S., An identity for a quasilinear ODE and its applications to the uniqueness of solutions of BVP, J. Math. Anal. Appl. 351 (2009), 206-217 . (2009) MR2472934
- Walter, W., Ordinary Differential Equations, Graduate Texts in Mathematics, 182, Springer, New York (1998) . Zbl1069.34095MR1629775
- Wang, H., On the existence of positive solutions for semilinear elliptic equations in the annulus, J. Differ. Equations 109 (1994), 1-7 . (1994) Zbl0798.34030MR1272398
- Yanagida, E., Sturmian theory for a class of nonlinear second-order differential equations, J. Math. Anal. Appl. 187 (1994), 650-662 . (1994) Zbl0816.34026MR1297048

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.