An overview on extremals and critical points of the Sobolev inequality in convex cones

Abstract: In this survey, we consider the sharp Sobolev inequality in convex cones. We also prove it by using the optimal transport technique. Then we present some results related to the Euler-Lagrange equation of the Sobolev inequality: the so-called critical p-Laplace equation. Finally, we discuss some stability results related to the Sobolev inequality.