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

Our aim is to investigate the existence of local maximizers for the singular Trudinger-Moser functional $$F_{\alpha }(u):={\int }_{\mathrm{\Omega }}\frac{e^{\alpha u^2}-1}{\mathrm{\mid }x{\mathrm{\mid }}^a}\mathrm{d}x,u\in W_0^{1,2}(\mathrm{\Omega }),$$ restricted to the manifold $\mathrm{\Sigma }=:\{u\in W_0^{1,2}(\mathrm{\Omega }):\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2=1\},$ where $a\in [0,2)$, $\alpha \ge 0$ and $\mathrm{\Omega }$ denotes a smooth bounded domain ${\mathbb{R}}^2$ containing the origin. Adimurthi and Sandeep (Nonlinear. Differ. Equ. Appl. $\mathbf{\text{13}}$, 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 $F_{\alpha }$ is bounded on $\mathrm{\Sigma }$ whenever $\alpha \le {\alpha }_a$. In addition, Csató and Roy (Calc. Var. Partial Differ. Equ., $\mathbf{\text{54}}$, 2015) were able to ensure the existence of maximizers for $F_{\alpha }$ on $\mathrm{\Sigma }$ when $\alpha \le {\alpha }_a$. In the supercritical regime $\alpha >{\alpha }_a$, the functional $F_{\alpha }$ becomes unbounded on $\mathrm{\Sigma }$. Nevertheless, we prove that $F_{\alpha }$ still possesses local maximizers on $\mathrm{\Sigma }$ beyond the critical threshold, at least for $\alpha >{\alpha }_a$ sufficiently close to ${\alpha }_a$. Our approach relies on a variational analysis near the set of maximizers associated with the critical parameter ${\alpha }_a$, together with a suitable local compactness argument. Our result improves and complements related findings due to Struwe (Ann. Inst. Henri Poincaré, Anal. Non Linéaire, $\mathbf{\text{5}}$, 1988).