A variational approach to uniqueness of ground states for certain quasilinear PDEs

We used a variational method based on optimal transport arguments to prove uniqueness of radial ground states for certain quasilinear elliptic equations, and we give the explicit expressions of the solutions. Our variational approach relies on a correspondence between the ground states of these equations and the equilibrium solutions of Fokker-Planck type equations. Our method also allows to identify all the optimal functions of many geometric inequalities, such as the Sobolev inequalities, the logarithmic Sobolev inequalities, and certain Gagliardo-Nirenberg inequalities.