Results on Parabolic Equations Related to Some Caffarelli-Kohn-Nirenberg Inequalities

Symmetry of Extremal Functions for the Caffarelli-kohn-nirenberg Inequalities

We study the symmetry property of extremal functions to a family of weighted Sobolev inequalities due to Caffarelli-Kohn-Nirenberg. By using the moving plane method, we prove that all non-radial extremal functions are axially symmetric with respect to a line passing through the origin.

Caffarelli – Kohn – Nirenberg inequalities with remainder terms $ Zhi - Qiang Wanga and Michel

Fractional Caffarelli-Kohn-Nirenberg inequalities

We establish a full range of Caffarelli-Kohn-Nirenberg inequalities and their variants for fractional Sobolev spaces.

Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities

We consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities and weighted logarithmic Hardy inequalities which have been obtained recently as a limit case of the first ones. We discuss the ranges of the parameters for which the optimal constants are achieved by extremal functions. The comparison of these optimal constants with the optimal constants of Gagliardo-Nirenberg interpo...

Perturbation Results of Critical Elliptic Equations of Caffarelli-kohn-nirenberg Type

We find for small ε positive solutions to the equation −div (|x| −2a ∇u) − λ |x| 2(1+a) u = 1 + εk(x) u p−1 |x| bp in R N , which branch off from the manifold of minimizers in the class of radial functions of the corresponding Caffarelli-Kohn-Nirenberg type inequality. Moreover, our analysis highlights the symmetry-breaking phenomenon in these inequalities, namely the existence of non-radial mi...