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

TL;DR

This paper investigates how the ground state energy of a critical quasilinear operator in ^N behaves under weak perturbations, providing asymptotic estimates for the lowest eigenvalue in different dimensional regimes.

Contribution

It offers new asymptotic formulas for the lowest eigenvalue of perturbed critical quasilinear operators, distinguishing between different dimensional cases.

Findings

Asymptotic behavior of the lowest eigenvalue for N > p^2

Asymptotic behavior of the lowest eigenvalue for N  p

Separate analysis for different dimensional regimes

Abstract

We consider a critical quasilinear operator $-\Delta_p u +V|u|^{p-2}u$ 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 $N\leq p^2$.