×

zbMATH — the first resource for mathematics

Uniqueness of positive solutions of nonlinear second-order equations. (English) Zbl 0857.34034
The author studies the question of uniqueness for positive solutions \(u\in C^2([-R,R])\) of the two-point boundary value problem (1) \(u''(t)+ f(|t|,u(t))=0\), \(-R<t<R\), \(u(\pm R)=0\), where \(R>0\) is fixed and the function \(f:[0,R]\times [0,\infty)\to\mathbb{R}^1\) is in \(\mathbb{C}^1 ([0,R]\times [0,\infty))\) and satisfies the following hypotheses: a) \(uf_u(t,u)> f(t,u)\) on \([0,R]\times [0,\infty)\); b) \(f_t(t,u)\leq 0\) on \([0,R]\times [0,\infty)\); c) there exists \(u>0\) such that \(f(R,u)\geq 0\).
The result of uniqueness for solutions of (1) is illustrated by examples for which existence is well known.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Robert Dalmasso, Uniqueness of positive and nonnegative solutions of nonlinear equations, Funkcial. Ekvac. 37 (1994), no. 3, 461 – 482. · Zbl 0819.34013
[2] D. G. de Figueiredo, P.-L. Lions, and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. (9) 61 (1982), no. 1, 41 – 63. · Zbl 0452.35030
[3] B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209 – 243. · Zbl 0425.35020
[4] Andrzej Granas, Ronald Guenther, and John Lee, Nonlinear boundary value problems for ordinary differential equations, Dissertationes Math. (Rozprawy Mat.) 244 (1985), 128. · Zbl 0615.34010
[5] Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. · Zbl 0123.21502
[6] M. A. Krasnosel\(^{\prime}\)skiń≠, Positive solutions of operator equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964.
[7] Man Kam Kwong, Uniqueness of positive solutions of \Delta \?-\?+\?^{\?}=0 in \?\(^{n}\), Arch. Rational Mech. Anal. 105 (1989), no. 3, 243 – 266. · Zbl 0676.35032
[8] P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), no. 4, 441 – 467. · Zbl 0511.35033
[9] Wei-Ming Ni and Roger D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of \Delta \?+\?(\?,\?)=0, Comm. Pure Appl. Math. 38 (1985), no. 1, 67 – 108. · Zbl 0581.35021
[10] Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. · Zbl 0609.58002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.