LAUSR

Let Ω be a smooth bounded domain in ℝ 2 , W 0 1,2 (Ω) be the standard Sobolev space. By the method of energy estimate developed by Malchiodi-Martinazzi ( J. Eur. Math. Soc. , 16 , 893–908 (2014)), Mancini-Martinazzi ( Calc. Var. Partial Differential Equations , 56 , 94 (2017)) and Mancini-Thizy ( J. Differential Equations , 266 , 1051–1072 (2019)), we reprove the results of Carleson-Chang ( Bull. Sci. Math. , 110 , 113–127 (1986)), Flucher ( Comment. Math. Helv. , 67 , 471–497 (1992)), Li ( Acta Math. Sin. Engl. Ser. , 22 , 545–550 (2006)) and Su ( J. Math. Inequal. , in press). Namely, for any real number α ≤ 1, the supremum $$\mathop {\sup}\limits_{v \in W_0^{1,2}\left({\rm{\Omega}} \right),\left\| {{\nabla _u}} \right\|_2^2 \le 4{\rm{\pi}}} \int_{\rm{\Omega}} {\left({{{\rm{e}}^{{v^2}}} - \alpha {v^2}} \right)dx}$$ sup v ∈ W 0 1 , 2 ( Ω ) , ‖ ∇ u ‖ 2 2 ≤ 4 π ∫ Ω ( e v 2 − α v 2 ) d x can be achieved by some function v ∈ W 0 1,2 (Ω) with ∥∇ v ∥ 2 2 ≤ 4π.