On the behavior of the ground state energy under weak perturbation of critical quasilinear operators in ${\mathbb{R}}^N$

We consider a critical quasilinear operator $-{\mathrm{\Delta }}_{p}u+V\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u$ in ${\mathbb{R}}^N$ perturbed by a weakly coupled potential. For $N>p$, we find the leading asymptotic of the lowest eigenvalue of such an operator in the weak coupling limit separately for $N>p^2$ and $N\le p^2$.