Variational and geometric methods for nonlinear differential equations

This thesis is devoted to the study of several problems arising in the field of nonlinear analysis. The work is divided in two parts: the first one concerns existence of oscillating solutions, in a suitable sense, for some nonlinear ODEs and PDEs, while the second one regards the study of qualitative properties, such as monotonicity and symmetry, for solutions to some elliptic problems in unbounded domains. Although the topics faced in this work can appear far away one from the other, the techniques employed in different chapters share several common features. In the firts part, the variational structure of the considered problems plays an essential role, and in particular we obtain existence of oscillating solutions by means of non-standard versions of the Nehari's method and of the Seifert's broken geodesics argument. In the second part, classical tools of geometric analysis, such as the moving planes method and the application of Liouville-type theorems, are used to prove 1-dimensional symmetry of solutions in different situations.