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 by Chen, Jiang, Lu and Zhu (2026), we resolve the endpoint cases: both thresholds for $p=2$ and the finite threshold for $2<p\le4$ are attained. We also prove compactness of centred radial extremals and one-sided value expansions governed by the selected endpoint $L^p$ mass. For $p>4$, we quantify how extremals spread out as $λ\to\infty$ by establishing the two-term expansion $S(λ,p)=4π+\mathcal C_pλ^{-2/(p-4)} +D_pλ^{-4/(p-4)}+o\bigl(λ^{-4/(p-4)}\bigr),$ Here $\mathcal C_p$ is the maximum of a limiting variational problem at unit $L^2$ mass. An explicit correction involving the sixth moment and the full $H^1$ normalization determines $D_p$ and selects the limiting maximizers. Rescaled centred radial extremals converge strongly to these profiles in $H^1(\mathbb{R}^2)$ along subsequences. For every fixed $λ>λ_*(4)$, we further establish $\lim_{p\downarrow4}(p-4)\log\bigl(S(λ,p)-4π\bigr) =2\log\frac{λ_{\mathrm{loc}}}λ<0,$ where $λ_{\mathrm{loc}}$ is the local critical penalty determined by the sharp Gagliardo–Nirenberg constant. The gradient energy has the same logarithmic rate, and the centred radial profiles, normalized in mass and gradient energy, converge along subsequences to sharp Gagliardo–Nirenberg optimizers. An exact transfer identity extends these endpoint and asymptotic results to every constant planar norm.