Maximizers for the Singular Trudinger-Moser functional beyond the critical regime

Abstract

Our aim is to investigate the existence of local maximizers for the singular Trudinger-Moser functional $$F_{\hat{I}}\pm (u):={\int }_{\hat{I}}\text{c◯}\frac{e^{\hat{I}\pm u^2}-1}{|x{|}^a}\mathrm{d}x,u\in W_0^{1,2}(\hat{I}\text{c◯}),$$ restricted to the manifold $\hat{I}\pounds =:\left\{u\in W_0^{1,2}(\hat{I}\text{c◯}):\Vert \nabla u{\Vert }_2=1\right\},$ where $a\in [0,2)$, $\hat{I}\pm \ge 0$ and $\hat{I}\text{c◯}$ denotes a smooth bounded domain ${\mathbb{R}}^2$ containing the origin. Adimurthi and Sandeep (Nonlinear. Differ. Equ. Appl. \textbf{13}, 2007) showed the following singular Trudinger-Moser type estimate \begin{equation}\nonumber \sup_{u \in W_0^{1,2}(Ω), \, \|\nabla u\|_{L^2} \le 1} F_α(u)<\infty\;\;\;\mbox{iff}\;\;α\leq α_a:=2π(2-a). \end{equation} In particular, the functional $F_{\hat{I}}\pm$ is bounded on $\hat{I}\pounds$ whenever $\hat{I}\pm \le \hat{I}{\pm }_a$. In addition, Csató and Roy (Calc. Var. Partial Differ. Equ., \textbf{54}, 2015) were able to ensure the existence of maximizers for $F_{\hat{I}}\pm$ on $\hat{I}\pounds$ when $\hat{I}\pm \le \hat{I}{\pm }_a$. In the supercritical regime $\hat{I}\pm >\hat{I}{\pm }_a$, the functional $F_{\hat{I}}\pm$ becomes unbounded on $\hat{I}\pounds$. Nevertheless, we prove that $F_{\hat{I}}\pm$ still possesses local maximizers on $\hat{I}\pounds$ beyond the critical threshold, at least for $\hat{I}\pm >\hat{I}{\pm }_a$ sufficiently close to $\hat{I}{\pm }_a$. Our approach relies on a variational analysis near the set of maximizers associated with the critical parameter $\hat{I}{\pm }_a$, together with a suitable local compactness argument. Our result improves and complements related findings due to Struwe (Ann. Inst. Henri Poincaré, Anal. Non Linéaire, \textbf{5}, 1988).