This PhD thesis collects many new results obtained via variational and topological techniques for different types of nonlinear partial differential equations which present quite challenging structures. The relevance of the problems handled here relies upon the fact that they outline the mathematical wording which many significant physical, chemical and biological phenomena are governed by. More precisely, not only we state new existence and multiplicity results for partial differential equations and/or systems which are semilinear, quasilinear (local and nonlocal) or of relativistic type, but also we provide several new abstract theorems which turn out that cannot be fit in proper tangible models or physical situations, but are of significant interest in current pure mathematical research.

Phd Thesis.