Existence and Nonexistence of Extremals for Trudinger–Moser Inequalities with \(Lᵖ\)-Type Perturbation on any Bounded Planar Domains

In this paper, we investigate the perturbed Trudinger–Moser inequalities as follows: [Formula: see text] where [Formula: see text] and [Formula: see text] is a bounded domain in [Formula: see text]. Our results demonstrate that there exists a threshold [Formula: see text] such that [Formula: see text] is attainable if [Formula: see text] but unattainable if [Formula: see text] when [Formula: see text]. For [Formula: see text], however, we show that [Formula: see text] is always attainable for any [Formula: see text]. These results are achieved through a refined blow-up analysis, which allows us to establish a sharp Dirichlet energy expansion formula for sequences of solutions to the corresponding Euler–Lagrange equations. The asymmetric nature of our problem poses significant challenges to our analysis. To address these, we will establish an appropriate comparison principle between radial and nonradial solutions of the associated Euler–Lagrange equations. Our study establishes a complete characterization of how [Formula: see text]-type perturbations influence the existence of extremals for critical Trudinger–Moser inequalities on any bounded planar domains. This extends the classical Brezis–Nirenberg problem framework to the two-dimensional setting.