Variational study of bifurcations in von Kármán equations

Abstract

For a class of nonlinear elliptic boundary value problems including the von Karman equations considered by D. M. Duc, N. L. Luc, L. Q. Nam, and T. T. Tuyen [Nonlinear Anal., 2003, 55: 951–968], we give a new proof of a corresponding theorem of three solutions via Morse theory instead of topological degree theory. Several bifurcation results for this class of boundary value problems are also obtained with Morse theory methods. In addition, for the von Karman equations studied by A. Borisovich and J. Janczewska [Abstr. Appl. Anal., 2005, 8: 889–899], we prove a few of bifurcation results under Dirichlet boundary conditions based on the second named author’s recent work about parameterized splitting theorems and bifurcations for potential operators.