| Abstract: |
| We consider the Trudinger-Moser inequality on a bounded domain with a scale parameter. We investigate the asymptotic behavior of maximizers attaining the inequality, with respect to the parameter. We first show that this behavior depends on the exponent in the Trudinger-Moser functional. When the exponent is large, an energy concentration phenomenon occurs, whereas when it is small, a vanishing phenomenon occurs.
We then focus on the vanishing phenomenon and establish the asymptotic expansion of the optimal constant, as well as the profile of maximizers exhibiting vanishing behavior. Furthermore, we show that, after a suitable transformation, the maximizers converge to a maximizer of a $L^2$-normalized variational problem. |
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