Uniqueness of positive solutions of nonlinear second-order equations
HTML articles powered by AMS MathViewer
- by Robert Dalmasso
- Proc. Amer. Math. Soc. 123 (1995), 3417-3424
- DOI: https://doi.org/10.1090/S0002-9939-1995-1301018-5
- PDF | Request permission
Abstract:
In this paper we study the uniqueness question of positive solutions of the two-point boundary value problem: $u''(t) + f(|t|,u(t)) = 0, - R < t < R,u( \pm R) = 0$ where $R > 0$ is fixed and $f:[0,R] \times [0,\infty ) \to \mathbb {R}$ is in ${C^1}([0,R] \times [0,\infty ))$. A uniqueness result is proved when f satisfies some appropriate conditions. Some examples illustrating our theorem are also given.References
- Robert Dalmasso, Uniqueness of positive and nonnegative solutions of nonlinear equations, Funkcial. Ekvac. 37 (1994), no. 3, 461–482. MR 1311555
- 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. MR 664341
- 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. MR 544879, DOI 10.1007/BF01221125
- Andrzej Granas, Ronald Guenther, and John Lee, Nonlinear boundary value problems for ordinary differential equations, Dissertationes Math. (Rozprawy Mat.) 244 (1985), 128. MR 808227
- Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
- M. A. Krasnosel′skiĭ, Positive solutions of operator equations, P. Noordhoff Ltd., Groningen, 1964. Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron. MR 0181881
- Man Kam Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\textbf {R}^n$, Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266. MR 969899, DOI 10.1007/BF00251502
- P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), no. 4, 441–467. MR 678562, DOI 10.1137/1024101
- Wei-Ming Ni and Roger D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u,r)=0$, Comm. Pure Appl. Math. 38 (1985), no. 1, 67–108. MR 768105, DOI 10.1002/cpa.3160380105
- 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. MR 845785, DOI 10.1090/cbms/065
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 3417-3424
- MSC: Primary 34B15; Secondary 34A12
- DOI: https://doi.org/10.1090/S0002-9939-1995-1301018-5
- MathSciNet review: 1301018