A symmetry theorem for variational problems.

*(English)*Zbl 0725.47013In the present paper nonlinear equations of the form
\[
(1)\quad Au=F(u),\quad u\in H,
\]
are considered. Here H is a Hilbert space, A: \(H\to H\) is a self-adjoint operator and F: \(H\to H\) is a potential operator, i.e. \(F=\text{grad} G\) with G: \(H\to R\). The author provides some assertions regarding the solvability of the system and the existence of special solutions respectively. Obviously (1) can be regarded as optimality condition of the variational problem
\[
(2)\quad \min \{f(u)=<Au,u>-G(u)| \quad u\in H\}.
\]
Accordingly the proof of the theorems are constructed in such a way that by a suitable decomposition of H in orthogonal subspaces (this is guaranteed by assumptions to the spectrum of A) and by monotony demands of F convexity properties in (2) and thus the existence of solutions of (1) can be pointed out.

Specifying A and F finally the author formulates special results for ordinary differential equation systems, for the Laplace equation, for discrete dynamical systems and for systems of nonlinear wave equations respectively.

Specifying A and F finally the author formulates special results for ordinary differential equation systems, for the Laplace equation, for discrete dynamical systems and for systems of nonlinear wave equations respectively.

Reviewer: J.Thierfelder (Ilmenau)

##### MSC:

47A50 | Equations and inequalities involving linear operators, with vector unknowns |

47B25 | Linear symmetric and selfadjoint operators (unbounded) |

49R50 | Variational methods for eigenvalues of operators (MSC2000) |

47N20 | Applications of operator theory to differential and integral equations |

##### Keywords:

saddle point; nonlinear equations; self-adjoint operator; potential operator; solvability; existence of special solutions; variational problem; ordinary differential equation systems; Laplace equation; discrete dynamical systems; systems of nonlinear wave equations
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\textit{M. Fečkan}, Nonlinear Anal., Theory Methods Appl. 16, No. 6, 499--506 (1991; Zbl 0725.47013)

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##### References:

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[2] | Amann, H., Saddle points and multiple solutions of differential equations, Math. Z., 169, 127-166, (1979) · Zbl 0414.47042 |

[3] | Ekeland, I.; Temam, R., Analyse convexe et problemes variationals, (1974), Dunod Paris |

[4] | Busenberg, S.; Fisher, D.; Martelli, M., Minimal periods of discrete and smooth orbits, Am. math. mon., 96, 5-17, (1989) · Zbl 0672.34034 |

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