Abstract
The existence of two intervals of positive real parameters λ for which the functional Φ + λΨ has three critical points, whose norms are uniformly bounded in respect to λ belonging to one of the two intervals, is established. As an example of an application to nonlinear differential problems, a two point boundary value problem is considered and multiplicity results are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Addou, I. (2000), Multiplicity results for classes of one-dimensional p-Laplacian boundary-value problems with cubic-like nonlinearities, Electron. J. Differ. Equ. (2000), 1–42.
Avery, R.I. and Henderson, J. (2000), Three symmetric positive solutions for a second-order boundary value problem, Appl. Math. Letters 13, 1–7.
Averna, D. and Bonanno, G., A three critical points theorem and its applications to the ordinary Dirichlet problem (in press).
Averna, D. and Bonanno, G., Three solutions for a quasilinear two point boundary value problem involving the one-dimensional p-Laplacian (in press).
Bonanno, G. (2000), Existence of three solutions for a two point boundary value problem, Appl. Math. Lett. 13, 53–56.
Bonanno, G. (2002), A minimax inequality and its applications to ordinary differential equations, J. Math. Anal. Appl. 270, 210–229.
Bonanno, G., Multiple solutions for a Neumann boundary value problem, J. Convex Nonlin. Anal. (in press).
Bonanno, G. and Candito, P. (2003), Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math. (Basel) 80, 424–429.
Bonanno, G. and Livrea, R. (2003), Multiplicity theorems for the Dirichlet problem involving the p-Laplacian, Nonlin. Anal. 54, 1–7.
Bonanno, G. (2003), Some remarks on a three critical points theorem, Nonlinear Anal. 54, 651–665.
Candito, P. (2000), Existence of three solutions for a nonautonomous two point boundary value problem, J. Math. Anal. Appl. 252, 532–537.
Dang, H., Schmitt, K. and Shivaji, R. (1996) On the number of solutions of boundary value problems involving the p-laplacian, Electron. J. Differ. Equ. 1996, 1–9.
Henderson, J. and Thompson, H.B. (2000), Existence of multiple solutions for second order boundary value problems, J. Differential Equations 166, 443–454.
Henderson, J. and Thompson, H.B. (2000), Multiple symmetric positive solutions for a second order boundary value problem, Proc. Am. Math. Soc. 128, 2373–2379.
Kong, L. and Wang, J. (2000), Multiple positive solutions for the one-dimensional p-Laplacian, Nonlin. Anal. 42, 1327–1333.
Livrea, R. (2002), Existence of three solutions for a quasilinear two point boundary value problem, Arch. Math. (Basel) 79, 288–298.
Marano, S.A. and Motreanu, D. (2002), On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems, Nonlin. Anal. 48, 37–52.
Pucci, P. and Serrin, J. (1985), A mountain pass theorem, J. Differential Equations 60, 142–149.
Ricceri, B. (2000), A general variational principle and some of its applications, J. Comput. Appl. Math. 113, 401–410.
Ricceri, B. (2000), On a three critical points theorem, Arch. Math. (Basel) 75, 220–226.
Ricceri, B., (2002), Three solutions for a Neumann problem, Topol. Methods Nonlinear Anal. 20, 275–281.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bonanno, G. A Critical Points Theorem and Nonlinear Differential Problems. Journal of Global Optimization 28, 249–258 (2004). https://doi.org/10.1023/B:JOGO.0000026447.51988.f6
Issue Date:
DOI: https://doi.org/10.1023/B:JOGO.0000026447.51988.f6