• nonlinear PDEs
• linking
• Kato-Ponce inequalities
• Dirac
• Coifman-Meyer
• profile decomposition
• rigidity
• solitons

The primary goal of the thesis is the study of the existence, uniqueness, regularity, and stability of solutions for several PDEs arising in physics such as the nonlinear Dirac equation.

The first goal is to present the notions of solitary wave and soliton as solutions of minimisation problems whose orbits satisfy precise properties. We show that, when the system preserves two physical quantities, solitons exist in a very general framework, provided that the functionals meet certain assumptions.

In the second chapter, we construct the topological degree in a finite-dimensional setting and show that we can extend it to Banach spaces through compact operators. The purpose is to use this topological tool to prove \textbf{deformation-type lemmas}, which, in turn, leads us to min-max theorems (e.g., the mountain pass or the saddle theorems.)
We mainly focus on linking-type theorems, which roughly assert that if we have two objects, N and C, that link, then the functional J admits a critical point at the level c, provided that J is "well-separated" between N and C, and satisfies certain regularity properties.

In the third chapter, we first present several tools in harmonic analysis, such as the Stein-Weiss interpolation theory, the Hardy-Littlewood maximal inequality, the Fefferman-Stein maximal inequality, and the Littlewood-Paley atomic decomposition.
Next, we employ all these tools to give a detailed proof of the well-known Coifman-Meyer multiplier theorem, which asserts that the bilinear operator of symbol a is bounded from L^p(\R^d) x L^q(\R^d) to L^r(\R^d).
We next employ these results to derive a Kato-Ponce-type inequality that involves the Riesz potential |D|^s. We also present a brief overview of the state-of-art of this kind of inequalities, pointing out the main gaps and the reason behind them.

In the second half of the thesis, we present some examples of PDEs arising from Physics, and we apply all the tools introduced so far to develop local/global well-posedness theories or investigate the existence of a solution under certain assumptions.

1) Schrödinger equation. We show that, in the energy subcritical case, we can always find hylomorphic solitons, if the nonlinearity is regular enough. We next deal with the critical case using a combination of two relatively recent tools: the profile decomposition and rigidity-type theorems.

2) Heat equation. We study the critical energy case using a slightly different profile decomposition and a rigidity theorem, which follows from uniqueness results fascinating enough by themselves. The somewhat simple structure of the heat equation allows us to provide more details about the method mentioned above.

3) Dirac equation. We investigate the existence of a radially symmetric solution of the Dirac equation with nonzero mass using linking-type theorems.

4) Burgers equation. We show that Kato-Ponce-type inequalities are significant when dealing with, e.g., hyperbolic PDEs for they can provide energy estimates using only integration by parts techniques.

In the appendix, we dedicate a chapter to illustrate Clifford algebras and their relation with higher-order Dirac matrices, which is a crucial step to attain a better understanding of the Dirac equation.