Existence and Nonexistence of Extremals for Trudinger-Moser inequalities with $L^p$ type perturbation on any bounded planar domains

In this study, we investigate the perturbed Trudinger-Moser inequalities as follows:$$S_{\mathrm{\Omega }}(\lambda ,p)=\underset{u\in H_0^1(\mathrm{\Omega }),\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2\left(\mathrm{\Omega }\right)}\le 1}{\sup }{\int }_{\mathrm{\Omega }}(e^{4\pi u^2}-\lambda \mathrm{\mid }u{\mathrm{\mid }}^p)dx,$$ where $1\le p<\mathrm{\infty }$ and $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^2$. Our results demonstrate that there exists a threshold ${\lambda }^{\ast }(p)>0$ such that $S_{\mathrm{\Omega }}(\lambda ,p)$ is attainable if $\lambda <{\lambda }^{\ast }(p)$, but unattainable if $\lambda >{\lambda }^{\ast }(p)$ when $p\in [1,2]$. For $p>2$, however, we show that $S_{\mathrm{\Omega }}(\lambda ,p)$ is always attainable for any $\lambda \in \mathbb{R}$. These results are achieved through a refined blow-up analysis, which allow 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 non-radial solutions of the associated Euler-Lagrange equations. Our study establishes a complete characterization of how $L^p$-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 settings.