Ljusternik-Schnirelman theory in partially ordered Hilbert spaces

We present several variants of Ljusternik-Schnirelman type theorems in partially ordered Hilbert spaces; which assert the locations of the critical points constructed by the minimax method in terms of the order structures. These results are applied to nonlinear Dirichlet boundary value problems to obtain the multiplicity of sign-changing solutions.

[1]  Zhi-Qiang Wang,et al.  Nonlinear boundary value problems with concave nonlinearities near the origin , 2001 .

[2]  John M. Neuberger,et al.  A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems , 1998 .

[3]  Helmut Hofer,et al.  A NOTE ON THE TOPOLOGICAL DEGREE AT A CRITICAL POINT OF MOUNTAINPASS-TYPE , 1984 .

[4]  A. Castro,et al.  A sign-changing solution for a superlinear Dirichlet problem with a reaction term nonzero at zero , 1997 .

[5]  G. Tarantello.  Nodal solutions of semilinear elliptic equations with critical exponent , 1992, Differential and Integral Equations.

[6]  A. Ambrosetti,et al.  Quasilinear equations with a multiple bifurcation , 1997, Differential and Integral Equations.

[7]  Shujie Li.  Some aspects of semilinear elliptic boundary value problem , 2000 .

[8]  Kung-Ching Chang,et al.  On the Morse indices of sign changing solutions of nonlinear elliptic problems , 2000 .

[9]  E. N. Dancer.  Positivity of Maps and Applications , 1995 .

[10]  T. Bartsch,et al.  On the existence of sign changing solutions for semilinear Dirichlet problems , 1996 .

[11]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[12]  M. A. Krasnoselʹskii.  Topological methods in the theory of nonlinear integral equations , 1968 .

[13]  E. N. Dancer.  MINIMAX METHODS IN CRITICAL POINT THEORY WITH APPLICATIONS TO DIFFERENTIAL EQUATIONS (CBMS Regional Conference Series in Mathematics 65) , 1987 .

[14]  M. Pino,et al.  Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking , 1996 .

[15]  Haim Brezis,et al.  Combined Effects of Concave and Convex Nonlinearities in Some Elliptic Problems , 1994 .

[16]  Goong Chen,et al.  A high-linking algorithm for sign-changing solutions of semilinear elliptic equations , 1999 .

[17]  Haim Brezis,et al.  Remarks on finding critical points , 1991 .

[18]  Zhi-Qiang Wang,et al.  Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems , 2000 .

[19]  R. Palais.  Lusternik-Schnirelman theory on Banach manifolds , 1966 .

[20]  Helmut Hofer,et al.  Variational and topological methods in partially ordered Hilbert spaces , 1982 .

[21]  H. Amann.  Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces , 1976 .

[22]  G Zhang,et al.  A VARIANT MOUNTAIN PASS LEMMA , 1983 .

[23]  E. N. Dancer,et al.  On sign-changing solutions of certain semilinear elliptic problems , 1995 .

[24]  E. N. Dancer.  On the indices of fixed points of mappings in cones and applications , 1983 .

[25]  Zhi-Qiang Wang.  On a superlinear elliptic equation , 1991 .

[26]  Kuang-Chao Chang.  In nite Dimensional Morse Theory and Multiple Solution Problems , 1992 .

[27]  J. G. Azorero,et al.  Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term , 1991 .

[28]  Yongxin Li,et al.  A Minimax Method for Finding Multiple Critical Points and Its Applications to Semilinear PDEs , 2001, SIAM J. Sci. Comput..

[29]  A. Ambrosetti,et al.  Multiplicity Results for Some Nonlinear Elliptic Equations , 1996 .

[30]  Hans-Peter Heinz,et al.  Free Ljusternik-Schnirelman theory and the bifurcation diagrams of certain singular nonlinear problems , 1987 .

[31]  M. Willem.  Minimax Theorems , 1997 .

[32]  P. Rabinowitz.  Minimax methods in critical point theory with applications to differential equations , 1986 .

[33]  David Clark,et al.  A Variant of the Lusternik-Schnirelman Theory , 1972 .

[34]  E. N. Dancer,et al.  On the profile of the changing sign mountain pass solutions for an elliptic problem , 2002 .

[35]  Jianxin Zhou,et al.  Algorithms and Visualization for solutions of nonlinear Elliptic equations , 2000, Int. J. Bifurc. Chaos.

[36]  Gabriella Tarantello,et al.  On semilinear elliptic equations with indefinite nonlinearities , 1993 .

[37]  An Abstract Critical Point Theorem and Applications , 1985 .

[38]  E. N. Dancer,et al.  The generalized Conley index and multiple solutions of semilinear elliptic problems , 1996 .

[39]  AZp index theory , 1990 .

[40]  S. Solimini,et al.  Some existence results for superlinear elliptic boundary value problems involving critical exponents , 1986 .

[41]  Existence of many sign-changing nonradial solutions for semilinear elliptic problems on thin annuli , 1999 .

[42]  Multiple and sign changing solutions of an elliptic eigenvalue problem with constraint , 2001 .

[43]  T. Bartsch.  Critical Point Theory on Partially Ordered Hilbert Spaces , 2001 .

[44]  E. N. Dancer,et al.  A Note on Multiple Solutions of Some Semilinear Elliptic Problems , 1997 .

[45]  P. Rabinowitz,et al.  Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems , 1977 .

[46]  J. M. Neuberger.  A numerical method for finding sign-changing solutions of superlinear Dirichlet problems , 1996 .

[47]  P. Rabinowitz.  Some Critical Point Theorems and Applications to Semilinear Elliptic Partial Differential Equations. , 1978 .

[48]  P. Rabinowitz,et al.  Dual variational methods in critical point theory and applications , 1973 .