Microscopic structures from reduction of continuum nonlinear problems

We present an application of the Amann--Zehnder exact finite reduction to a class of nonlinear perturbations of elliptic elasto--static problems. We propose the existence of minmax solutions by applying Ljusternik--Schnirelmann theory to a finite dimensional variational formulation of the problem, based on a suitable spectral cut--off. As a by--product, with a choice of fit variables, we establish a variational equivalence between the above spectral finite description and a discrete mechanical model. By doing so, we decrypt the abstract information encoded in the AZ reduction and give rise to a concrete and finite description of the continuous problem.