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

Satoshi Tanaka

Mathematica Bohemica (2010)

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

Abstract

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The two-point boundary value problem u ' ' + h ( x ) u p = 0 , a < x < b , u ( a ) = u ( b ) = 0 is considered, where p > 1 , h ∈ C 1 [ 0 , 1 ] and h ( x ) > 0 for a ≤ x ≤ 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.

How to cite

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Tanaka, 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

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  10. Ni, W.-M., Nussbaum, R. D., Uniqueness and nonuniqueness for positive radial solutions of Δ u + f ( u , r ) = 0 , Comm. Pure Appl. Math. 38 (1985), 67-108 . (1985) Zbl0581.35021MR0768105
  11. 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
  12. 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
  13. Walter, W., Ordinary Differential Equations, Graduate Texts in Mathematics, 182, Springer, New York (1998) . Zbl1069.34095MR1629775
  14. 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
  15. Yanagida, E., Sturmian theory for a class of nonlinear second-order differential equations, J. Math. Anal. Appl. 187 (1994), 650-662 . (1994) Zbl0816.34026MR1297048

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