On some nonlocal elliptic problems

Résumé

This work is devoted to study some nonlocal elliptic problems involving the so-called fractional Laplacian, $(-\Delta)^{\alpha/2}$. In order to deal with these problems, several classical methods and techniques have been generalized to our context. Among these techniques we include the moving planes method, concentration compactness principle and a Pohozaev-type equation. This work is divided as follows: - Chapter 1: Introduction to the fractional Laplacian operator. - Chapter 2: A concave-convex elliptic problem involving the fractional Laplacian - Chapter 3: On some critical problems for the fractional Laplacian operator - Chapter 4: Perturbations of a critical fractional equation