• Nonlinear eigenvalues
• p -Laplace operator
• variational convergence

The purpose of this thesis is to investigate the convergence of critical
points of certain class functionals. These critical points are solutions of a class of quasi-linear boundary value problems and are obtained by means
of topological and variational methods of the theory of partial differential
equations. In particular, minimax methods and topological index theory are
used to define critical points, while the theory of Γ-convergence is employed
to obtain their convergence.
The first chapter collects some preliminary results. We present the main
ideas regarding the direct method of the calculus of variations, which can be
seen as a classic starting point for our work, then we define Γ-convergence
and show a fundamental result that concerns the convergence of minima
for Γ-converging functionals. The chapter ends with a description of some
variational methods and tools, such as Palais-Smale condition, the deformation lemma
and minimax methods.
In the second chapter we start an investigation of the problem of eigenvalues
for the p-Laplacian, and we define the tools that we use in the rest of the
work: we present index theory and define the Krasnoselskii genus and the
Z_2 - cohomological index, which are used to define several
sequences of eigenvalues for the p-Laplacian as minimax critical values. In
the last section, Hausdorff distance on compact sets is investigated.
In the third chapter we develop the main ideas of the work. Given a func-
tional f on a Banach space V we define a new functional in the space of
compact subsets of V as F(K) = sup_K f and we show that under proper
hypothesis if a sequence of functionals f_k Γ-converges to f_0 then the related
sequence of functionals F_k Γ-converges to F_0 with respect to the Hausdorff
distance. As a corollary, we obtain the convergence of sequences of eigen-
values for the p-Laplacian when p → ∞ as a convergence of minima in the
framework of variational convergence of functionals.