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 result related to the Sobolev inequality.