Large-amplitude periodic oscillations in simple and complex mechanical system: outgrowths from nonlinear analysis.

*(English)*Zbl 1117.35025The author starts his paper with a detailed review of the existing literature on semilinear elliptic equations or systems of the type \(\Delta u+f(u)=h(x)\), posed in a smooth and simply connected domain \(\Omega \) of \({\mathbb R}^{3}\), with homogeneous Dirichlet boundary conditions. Then the author describes some possible applications of the existence results to the study of mechanical systems which can be associated to a nonlinear Hamiltonian.

The author starts his review with the results from A. Ambrosetti and G. Prodi [Ann. Mat. Pura Appl. (4) 93, 231–246 (1972; Zbl 0288.35020)] and gathers existence results for such systems depending on the nonlinearity \(f\) and its properties. It is known that such systems may have multiple solutions, the number of which depends on the properties of the nonlinearity f with respect to the eigenvalues of the Laplace operator. In a short third part, the author presents some available existence results for nonlinear wave equations which are 1D in space. In the fourth until sixth parts of the paper, he applies these existence results and qualitative properties of the possibly multiple solutions to mechanical systems. The author here starts with the single mass-spring problem. Then he moves to the modelisation of suspension bridges considered as beams. Although the paper mainly focuses on semilinear equations, the author makes some analysis of systems or of the \(p\)-Laplacian operator within this context.

Throughout the whole work, the author raises some open problems concerning possible extensions of available results within this field. The author refers to a long list of references for the proof of the results, references which will be useful for people involved in this field.

The author starts his review with the results from A. Ambrosetti and G. Prodi [Ann. Mat. Pura Appl. (4) 93, 231–246 (1972; Zbl 0288.35020)] and gathers existence results for such systems depending on the nonlinearity \(f\) and its properties. It is known that such systems may have multiple solutions, the number of which depends on the properties of the nonlinearity f with respect to the eigenvalues of the Laplace operator. In a short third part, the author presents some available existence results for nonlinear wave equations which are 1D in space. In the fourth until sixth parts of the paper, he applies these existence results and qualitative properties of the possibly multiple solutions to mechanical systems. The author here starts with the single mass-spring problem. Then he moves to the modelisation of suspension bridges considered as beams. Although the paper mainly focuses on semilinear equations, the author makes some analysis of systems or of the \(p\)-Laplacian operator within this context.

Throughout the whole work, the author raises some open problems concerning possible extensions of available results within this field. The author refers to a long list of references for the proof of the results, references which will be useful for people involved in this field.

Reviewer: Alain Brillard (Riedisheim)

##### MSC:

35J60 | Nonlinear elliptic equations |

74H45 | Vibrations in dynamical problems in solid mechanics |

70K42 | Equilibria and periodic trajectories for nonlinear problems in mechanics |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |