Some applications of the generalized Morse-Conley index.

*(English)*Zbl 0656.58006
Conf. Semin. Mat. Univ. Bari 218, 32 p. (1987).

The author proves some multiplicity results for partial and ordinary differential equations by means of his generalization of Morse-Conley index to the infinite dimensional case. In the first two sections the basic notions and results of this theory are reviewed. In the third section some applications to partial differential equations are given. In particular it is proved the existence of a nontrivial solution u of the problem \(u\in H^ 1_ 0(\Omega)\), \(-\Delta u=G'(u)\), where G has superquadratic growth at infinity, \(G'(0)=0\) and G”(0) is not an eigenvalue of \(-\Delta\). In the last section the equivariant case is treated with applications to ordinary differential systems. In particular it is shown that the ordinary system \(q''+\text{grad}_ qV(t,q)=0\) has infinitely many T-periodic solutions q, provided that V is T-periodic, has superquadratic growth at infinity and satisfies a suitable symmetry assumption in the space variables.

Reviewer: M.Degiovanni

##### MSC:

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

58J99 | Partial differential equations on manifolds; differential operators |

37G99 | Local and nonlocal bifurcation theory for dynamical systems |