Threshold Attainment and Sharp Asymptotics for $L^p$-Perturbed Moser--Trudinger Functionals on $\mathbb{R}^2$

In this paper, we study the critical Moser--Trudinger supremum \[ S(λ,p) = \sup_{\substack{ u\in H^1(\mathbb{R}^2)\\ \int_{\mathbb{R}^2} (|\nabla u|^2+u^2)\,dx\le 1 }} \int_{\mathbb{R}^2} \left(e^{4πu^2}-1-λ|u|^p\right)\,dx, \qquad p\ge 2. \] Building on the classification away from the thresholds b