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Preface: Applications of mathematical analysis to problems in theoretical physics.

We study the field equation $$-Delta u+V(x)u+arepsilon^r(-Delta_pu+W'(u))=mu u$$ on $mathbb R^n$, with $arepsilon$ positive parameter. The function $W$ is singular in a point and so the configurations are characterized by a topological invariant: the topological charge. By a min-max method, for $arepsilon$ sufficiently small, there exists a finite number of solutions $(mu(arepsilon),u(arepsilon))$ of the eigenvalue problem for any given charge $qin{mathbb Z}setminus{0}$.

Hamiltonian systems of ordinary differential equations model the motion of a discrete mechanical system when no frictional forces are present. A basic property of such systems is that energy is conserved. Therefore solutions of Hamiltonian systems lie on surfaces of fixed energy. The main result of this paper is a fairly general criterion for such a surface to possess a periodic solution.

Questions of existence of periodic solutions for classical mechanical systems have a long history. The development of the nonlinear functional analysis has provided powerful new tools and renewed interest in these problems. This paper considers the Hamiltonian system under certain circumstances.

This paper uses Conley's index to study the cortical points of a function f on a finite dimensional sphere in presence of a symmetry group. The authors prove a theorem which leads to a lower bound on the number of critical points of f when the group is finite, even if the action is not free. This investigation has been motivated by the following bifurcation problem Au = lambda, where A is a variational G-equivariant operator. An estimated on the number of 'branches' bifurcating from an eigenvalue of A'(0) is given. (Author)

This paper studies harmonic maps when omega is the two dimensional disk and M = sub n. In this situation, given a smooth function gamma from curly d omega to sub n we prove that if gamma is not constant, there exist two harmonic functions u such that u/curly d omega = gamma.

The question of existence and the number of periodic solutions (normal modes) for a classical mechanical system is a problem as old as the field of analytical mechanics itself. The development of the nonlinear functional analysis has renewed interest in these problems. This paper considers a mechanical system which is constrained in a potential well.