Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof

Abstract We prove the first genuine “partial differential equation” result on a conjecture concerning the number of solutions of second-order elliptic boundary value problems with a nonlinearity which grows superlinearly at +∞. The proof makes massive use of computer assistance: After approximate solutions have been computed by a numerical mountain pass algorithm, combined with a Newton iteration to improve accuracy, a fixed point argument is used to show the existence of exact solutions close to the approximations.

[1]  Alan C. Lazer,et al.  Large-Amplitude Periodic Oscillations in Suspension Bridges: Some New Connections with Nonlinear Analysis , 1990, SIAM Rev..

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

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

[4]  G. Corliss,et al.  C-Xsc: A C++ Class Library for Extended Scientific Computing , 1993 .

[5]  M. Kwong.  On the unboundedness of the number of solutions of a Dirichlet problem , 1992 .

[6]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[7]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[8]  Alan C. Lazer,et al.  A symmetry theorem and applications to nonlinear partial differential equations , 1988 .

[9]  A. Castro,et al.  The Lazer Mckenna Conjecture for Radial Solutions in the R(N) Ball , 1993 .

[10]  Michael Plum.  Bounds for eigenvalues of second-order elliptic differential operators , 1991 .

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

[12]  Michael Plum,et al.  Computer-assisted enclosure methods for elliptic differential equations , 2001 .

[13]  Michael Plum,et al.  Explicit H2-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems , 1992 .

[14]  Alan C. Lazer,et al.  On the number of solutions of a nonlinear Dirichlet problem , 1981 .

[15]  E. N. Dancer.  A counterexample to the Lazer-McKenna conjecture , 1988 .

[16]  A. Ambrosetti,et al.  On the inversion of some differentiable mappings with singularities between Banach spaces , 1972 .

[17]  S. Solimini,et al.  On a class of superlinear sturm Liouville problems with arbitrarily many solutions , 1986 .

[18]  F. W. Warner,et al.  Remarks on some quasilinear elliptic equations , 1975 .

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

[20]  Y. Choi,et al.  A mountain pass method for the numerical solution of semilinear wave equations , 1993 .

[21]  Peter Hess,et al.  A multiplicity result for a class of elliptic boundary value problems , 1979, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[22]  A. Castro,et al.  Radially symmetric solutions to a superlinear Dirichlet problem in a ball with jumping nonlinearities , 1987 .

[23]  Y. Choi,et al.  A mountain pass method for the numerical solution of semilinear elliptic problems , 1993 .

[24]  D. Costa,et al.  The exact number of solutions for a class of ordinary differential equations through Morse Index computation , 1992 .

[25]  P. J. McKenna,et al.  On a conjecture related to the number of solutions of a nonlinear Dirichlet problem , 1983, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[26]  P. N. Srikanth,et al.  Multiplicity results for an ODE problem with even nonlinearity , 1988 .