Rocky Mountain Journal of Mathematics

A Mountain Pass Theorem for a Suitable Class of Functions

Diego Averna and Gabriele Bonanno

Full-text: Open access

Article information

Rocky Mountain J. Math., Volume 39, Number 3 (2009), 707-727.

First available in Project Euclid: 18 May 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47J30: Variational methods [See also 58Exx] 58E30: Variational principles 49J40: Variational methods including variational inequalities [See also 47J20] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.) 34B15: Nonlinear boundary value problems

Palais-Smale condition mountain pass critical points three solutions two-point boundary value problem


Averna, Diego; Bonanno, Gabriele. A Mountain Pass Theorem for a Suitable Class of Functions. Rocky Mountain J. Math. 39 (2009), no. 3, 707--727. doi:10.1216/RMJ-2009-39-3-707.

Export citation


  • A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Anal. 14 (1973), 349-381.
  • D. Averna and G. Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem, Topol. Methods Nonlinear Anal. 22 (2003), 93-104.
  • --------, Three solutions for a quasilinear two point boundary value problem involving the one-dimensional $p$-Laplacian, Proc. Edinburgh Math. Soc. 47 (2004), 257-270.
  • R.I. Avery and J. Henderson, Three symmetric positive solutions for a second-order boundary value problem, Appl. Math. Letters 13 (2000), 1-7.
  • G. Bonanno, Multiple critical points theorems without the Palais-Smale condition, J. Math. Anal. Appl. 299 (2004), 600-614.
  • --------, Some remarks on a three critical points theorem, Nonlinear Anal. 54 (2003), 651-665.
  • --------, A critical points theorem and nonlinear differential problems, J. Global Optimization 28 (2004), 249-258.
  • G. Bonanno and P. Candito, Three solutions to a Neumann problem for elliptic equations involving the $p$-Laplacian, Arch. Math. 80 (2003), 424-429.
  • G. Bonanno and R. Livrea, Multiplicity theorems for the Dirichlet problem involving the $p$-Laplacian, Nonlinear Anal. 54 (2003), 1-7.
  • G. Bonanno and D. O'Regan, A boundary value problem on the half-line via critical point methods, Dynamic Syst. Appl. 15 (2006), 395-408.
  • K.-C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129.
  • F.H. Clarke, Optimization and nonsmooth analysis, Classics Appl. Math. 5, Soc. Indust. Appl. Mathematics, Philadelphia, 1990.
  • J. Henderson and H.B. Thompson, Existence of multiple solutions for second order boundary value problems, J. Differential Equations 166 (2000), 443-454.
  • --------, Multiple symmetric positive solutions for a second order boundary value problem, Proc. Amer. Math. Soc. 128 (2000), 2373-2379.
  • S.A. Marano and D. Motreanu, On a three critical points theorem for non differentiable functions and applications to non linear boundary value problems, Nonlinear Anal. 48 (2002), 37-52.
  • D. Motreanu and V. Rǎdulescu, Variational and non-variational methods in nonlinear analysis and boundary value problems, Kluwer, Dordrecht, 2003.
  • P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations 63 (1985), 142-149.
  • P.H. Rabinowitz, Minimax methods in critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Math. 65, American Mathematical Society, Providence, RI, 1986.
  • B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 901-410.
  • --------, On a three critical points theorem, Arch. Math. 75 (2000), 220-226.
  • R. Salvati, Multiple solutions for a mixed boundary value problem, Math. Sci. Res. J. 7 (2003), 275-983.
  • E. Zeidler, Nonlinear functional analysis and its applications, Vol. III., Springer-Verlag, Berlin, 1985.