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

Abstract

In this study, we investigate the perturbed Trudinger-Moser inequalities as follows:\[ S_Ω(λ,p)=\sup_{u\in H_{0}^{1}(Ω),\Vert\nabla u\Vert _{L^{2}\left( Ω\right) }\leq 1}\int_Ω\left( e^{4πu^{2}}-λ|u|^{p}\right) dx, \] where $1\le p<\infty$ and $\Omega$ is a bounded domain in ${\mathbb{R}}^2$. Our results demonstrate that there exists a threshold ${\lambda }^{\ast }(p)>0$ such that $S_{\Omega }(\lambda ,p)$ 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 $S_{\Omega }(\lambda ,p)$ 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.

52Pages