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Existence of three solutions for a Kirchhoff-type boundary-value problem. (English) Zbl 1234.34018

The authors consider the following boundary-value problem of Kirchhoff-type \[ {-K}\left( {\int_{a}^{b}|u^{\prime}(x)|^2\, dx}\right) {u^{\prime \prime }=\lambda f(x,u),}\;{u(a)=u(b)=0}, \] where \(\lambda >0,\) \(K:\left[ 0,+\infty \right) \rightarrow \mathbb R\) is continuous and \(f:\left[ a,b\right] \times \mathbb R\rightarrow \mathbb R\) is a Carathéodory function, which functions are also subject to some conditions. Using a three critical point theorem by Bonanno, they establish the existence of two intervals of positive real parameters \(\lambda \) for which the problem under consideration admits three weak solutions whose norms are uniformly bounded with respect to \(\lambda \) belonging to one of the two intervals.

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E30 Variational principles in infinite-dimensional spaces
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