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Existence of three solutions for a two point boundary value problem. (English) Zbl 1009.34019
From the introduction: The autonomous ordinary Dirichlet problem $$u''+ \lambda f(u)= 0,\quad u(0)= u(1)= 0,\tag 1$$ is considered, where $\lambda$ is a positive parameter and $f: \bbfR\to\bbfR$ is a continuous function. Under a completely novel assumption on the function $\xi\to \int^\xi_ 0 f(t) dt$, the existence of an open interval $\Lambda\subseteq[0,+\infty[$ is proved such that, for every $\lambda\in \Lambda$, problem (1) has at least three classical solutions.

34B15Nonlinear boundary value problems for ODE
47J30Variational methods (nonlinear operator equations)
58E05Abstract critical point theory
Full Text: DOI
[1] B. Ricceri, On a three critical points theorem (preprint). · Zbl 0979.35040
[2] Gelfand, I. M.: Some problems in the theory of quasilinear equations. Amer. math. Soc. translations 29, 295-381 (1963) · Zbl 0127.04901
[3] Korman, P.; Ouyang, T.: Exact multiplicity results for a class of boundary-value problems with cubic nonlinearities. J. math. Anal. appl. 194, 328-341 (1995) · Zbl 0837.34033
[4] Korman, P.; Ouyang, T.: Exact multiplicity results for two classes of boundary value problem. Diff. integral eqns. 6, 1507-1517 (1993) · Zbl 0780.34013
[5] B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue problems, Mathl. Comput. Modelling, Special Issue on ”Advanced topics in nonlinear operator theory” (Edited by R.P. Agarwal and D. O’Regan) (to appear).