×

zbMATH — the first resource for mathematics

On the global set of solutions of a nonlinear ODE: theoretical and numerical description. (English) Zbl 0597.34014
Nonlinear differential equations of the type \(-u''=f(u)+\lambda g(x)\) in (0,1), \(u(0)=u(1)=0\) are studied. Here f is a convex, nondecreasing function with \(f(0)=0\), \(f'(0)=0\), \(\lim_{s\to \pm \infty}f(s)/s=\infty\) and g is a positive function in (0,1). In the particular case, \(f(u)=u| u|^{p-1}\) and g(x)\(\equiv 1\), a complete global description of the set of solutions is given exhibiting an infinite number of bifurcation points and turning points. Some of the results regarding the positive solutions are shown to extend for a similar semilinear elliptic problem in higher dimensions.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
34C99 Qualitative theory for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Khodja, A, Une revue et quelques compléments sur la Détermination du nombre des solutions de certains problèmes elliptiques semi-linéaires, Thése de troisième cycle, (1983), Paris VI
[2] Bahri, A; Berestycki, H, A perturbation method in critical point theory and applications, Trans. amer. math. soc., 207, 1-32, (1981) · Zbl 0476.35030
[3] \scP. Baras and M. Pierre, to appear.
[4] Berestycki, H, Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. funct. anal., 40, 1-29, (1981) · Zbl 0452.35038
[5] Crandall, M.G; Rabinowitz, P.H, Bifurcation from simple eigenvalues, J. funct. anal., 8, 321-340, (1981) · Zbl 0219.46015
[6] Crandall, M.G; Rabinowitz, P.H, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. rational mech. anal., 52, 161-180, (1973) · Zbl 0275.47044
[7] Crandall, M.G; Rabinowitz, P.H, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. rational mech. anal., 58, 207-218, (1975) · Zbl 0309.35057
[8] De Figueiredo, D.G; Lions, P.L; Nussbaum, R.D, A priori estimates and existence of positive solutions of semi-linear equations, J. math. pures appl., 61, 41-63, (1982) · Zbl 0452.35030
[9] De Mottoni, P; Tesai, A, On the solution of a class of nonlinear Sturm Liouville problems, SIAM J. math. anal., 9, 1020-1029, (1978) · Zbl 0399.34017
[10] Ehrmann, H, Über die existenz der Lösungen von randwertanfgaben bei gawöhnlicher nichlinearen differentialgleichungen zweiter ordnung, Math. ann., 134, 167-194, (1957) · Zbl 0078.07801
[11] Fučik, S; Lovicar, V, Periodic of the equations x″ (t) + g (x, t) = p (t), C̆asopis Pĕst. mat., 100, 160-175, (1975) · Zbl 0319.34038
[12] Glowinski, R; Keller, H.B; Reinhart, L, Continuation conjugate gradient methods for the least square solution of nonlinear boundary value problems, INRIA report no. 141, (1982) · Zbl 0589.65075
[13] Keller, H.B, Numerical solution of bifurcation and nonlinear eigenvalue problems, () · Zbl 0581.65043
[14] Laetsch, T, The number of solutions of a nonlinear two point boundary value problem, Indiana univ. math. J., 20, 1-13, (1970) · Zbl 0215.14602
[15] Lions, J.L, Contrôle des systèmes distribués singulièrs, (1983), Dunod Paris · Zbl 0514.93001
[16] Lions, P.L, On the existence of positive solutions of semilinear elliptic equations, SIAM rev., 24, 441-467, (1982) · Zbl 0511.35033
[17] \scH. O. Peitgen and K. Schmitt, Global analysis of a two parameter boundary value problems, Trans. Amer. Math. Soc., in press. · Zbl 0543.35039
[18] Pohazaev, S.I, Eigenfunctions of δu + λf (u) = 0, Soviet math. dokl., 6, 1408-1411, (1965) · Zbl 0141.30202
[19] Rabinowitz, P.H, Variational methods for nonlinear eigenvalue problems, () · Zbl 0212.16504
[20] Reinhart, L, Sur la Résolution numérique de problèmes aux limites nonlinéaires par des Méthods de continuation, Thése de troisième cycle, (1980), Paris VI
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.