Perturbation results of critical elliptic equations of Caffarelli–Kohn–Nirenberg type

Abstract

We find for small ε positive solutions to the equation

$$\text{−}\text{div}\text{(|x|}{}^{\mathrm{-2a}}\text{∇}\text{u)−}\text{λ}\text{|x|}{}^{\mathrm{2(1+a)}}\text{u=(1+εk(x))}\text{u}{}^{\mathrm{p-1}}\text{|x|}{}^{\mathrm{bp}}$$

in $\text{R}{}^{\mathrm{N}}$, 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 minimizers.