Let
$ \lambda_p(\Omega) = \inf\limits_{u\in W_0^{1, 2}(\Omega), u\not\equiv0} \frac{\|\nabla u\|_2^2}{\|u\|_p^2}. $
We prove that for any
$ \begin{equation*} \sup\limits_{u\in W_0^{1, 2}(\Omega), \, \| \nabla u\|_{2}^2\leq4 \pi}\int_{\Omega}e^{ u^2 (1+\tau\|u\|_p^2)}dx, \end{equation*} $
can not be achieved by any
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