| Title:
|
On the uniqueness of positive solutions for two-point boundary value problems of Emden-Fowler differential equations (English) |
| Author:
|
Tanaka, Satoshi |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
135 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
189-198 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The two-point boundary value problem \[ u'' + h(x) u^p = 0, \quad a < x < b, \qquad u(a) = u(b) = 0 \] 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. (English) |
| Keyword:
|
uniqueness |
| Keyword:
|
positive solution |
| Keyword:
|
two-point boundary value problem |
| Keyword:
|
Emden-Fowler equation |
| MSC:
|
34B15 |
| idZBL:
|
Zbl 1224.34075 |
| idMR:
|
MR2723086 |
| DOI:
|
10.21136/MB.2010.140696 |
| . |
| Date available:
|
2010-07-20T18:36:50Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140696 |
| . |
| Reference:
|
[1] Coffman, C. V.: On the positive solutions of boundary-value problems for a class of nonlinear differential equations.J. Differ. Equations 3 (1967), 92-111. MR 0204755, 10.1016/0022-0396(67)90009-5 |
| Reference:
|
[2] Dalmasso, R.: Uniqueness of positive solutions of nonlinear second-order equations.Proc. Amer. Math. Soc. 123 (1995), 3417-3424 . Zbl 0857.34034, MR 1301018, 10.1090/S0002-9939-1995-1301018-5 |
| Reference:
|
[3] Korman, P.: On the multiplicity of solutions of semilinear equations.Math. Nachr. 229 (2001), 119-127 . Zbl 0999.35028, MR 1855158 |
| Reference:
|
[4] Kwong, M. K.: On the Kolodner-Coffman method for the uniqueness problem of Emden-Fowler BVP.Z. Angew. Math. Phys. 41 (1990), 79-104 . Zbl 0708.34026, MR 1036511 |
| Reference:
|
[5] Kwong, M. K.: Uniqueness results for Emden-Fowler boundary value problems.Nonlinear Anal. 16 (1991) 435-454 . MR 1093379 |
| Reference:
|
[6] Moore, R., Nehari, Z.: Nonoscillation theorems for a class of nonlinear differential equations.Trans. Amer. Math. Soc. 93 (1959), 30-52 . MR 0111897 |
| Reference:
|
[7] Moroney, R. M.: Note on a theorem of Nehari.Proc. Amer. Math. Soc. 13 (1962) 407-410 . Zbl 0115.30601, MR 0148983 |
| Reference:
|
[8] Naito, M., Naito, Y.: Solutions with prescribed numbers of zeros for nonlinear second order differential equations.Funkcial. Ekvac. 37 (1994) 505-520 . Zbl 0820.34019, MR 1311557 |
| Reference:
|
[9] Naito, Y.: Uniqueness of positive solutions of quasilinear differential equations.Differ. Int. Equations 8 (1995), 1813-1822 . Zbl 0831.34028, MR 1347982 |
| Reference:
|
[10] 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 . Zbl 0581.35021, MR 0768105 |
| Reference:
|
[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. Zbl 1212.34040, MR 2282828 |
| Reference:
|
[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 . MR 2472934 |
| Reference:
|
[13] Walter, W.: Ordinary Differential Equations.Graduate Texts in Mathematics, 182, Springer, New York (1998) . Zbl 1069.34095, MR 1629775 |
| Reference:
|
[14] Wang, H.: On the existence of positive solutions for semilinear elliptic equations in the annulus.J. Differ. Equations 109 (1994), 1-7 . Zbl 0798.34030, MR 1272398 |
| Reference:
|
[15] Yanagida, E.: Sturmian theory for a class of nonlinear second-order differential equations.J. Math. Anal. Appl. 187 (1994), 650-662 . Zbl 0816.34026, MR 1297048 |
| . |
