This book deals with a systematic study of a dynamical system approach to investigate the
symmetrization and stabilization properties of nonnegative solutions of nonlinear elliptic
problems in asymptotically symmetric unbounded domains. The usage of infinite dimensional
dynamical systems methods for elliptic problems in unbounded domains as well as finite
dimensional reduction of their dynamics requires new ideas and tools. To this end both a
trajectory dynamical systems approach and new Liouville type results for the solutions of some
class of elliptic equations are used. The work also uses symmetry and monotonicity results for
nonnegative solutions in order to characterize an asymptotic profile of solutions and compares
a pure elliptic partial differential equations approach and a dynamical systems approach. The
new results obtained will be particularly useful for mathematical biologists.
