  In this paper we give an unified approach to some questions arising in
different fields of nonlinear analysis, namely: (a) the study of the structure
of the Fu\v{c}\'\i k spectrum and (b) possible variants and extensions of the
monotonicity formula by Alt--Caffarelli--Friedman \cite{acf}. In the first part
of the paper we present a class of optimal partition problems involving the
first eigenvalue of the Laplace operator. Beside establishing the existence of
the optimal partition, we develop a theory for the extremality conditions and
the regularity of minimizers. As a first application of this approach, we give
a new variational characterization of the first curve of the Fu\v{c}\'\i k
spectrum for the Laplacian, promptly adapted to more general operators. In the
second part we prove a monotonicity formula in the case of many subharmonic
components and we give an extension to solutions of a class of
reaction--diffusion equation, providing some Liouville--type theorems.
