Ground states in Mathematical Physics

Abstract

The main aim of this bachelor's thesis is to introduce the concept of a ground state and prove its existence for the nonlinear diffusion equation $$ -\Delta u + a u = b |u|^\alpha u \ \ \text{in} \ \mathbb{R}^N, $$ as well as, for the equation with a fractional diffusion: $$ \left(-\Delta\right)^s u + a u = b |u|^\alpha u \ \ \text{in} \ \mathbb{R}^N. $$ A ground state of a problem is, roughly speaking, a solution that minimises the total energy. \newline \par We will begin by introducing the Sobolev spaces and presenting some important results. We then define the concept of ground state and proceed to prove its existence for a nonlinear diffusion equation. Finally, we present the fractional Laplacian operator $ (-\Delta)^s $ and the fractional Sobolev space and explain the parallelisms when proving the existence of a ground state between the non-fractional and the fractional case.