On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms

Let $ \Omega $ be the smooth bounded domian in $ \mathbb{R}^2 $, $ W_0^{1, 2}(\Omega) $ be the standard Sobolev space. We concern a Trudinger-Moser inequality involving $ L^p $ norms. For any $ p>1 $, denote

$ \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 $ p>1 $ and any $ 0\leq\tau<\lambda_p $, there exists a positive real number $ \tau^\ast $ such that if $ \tau^\ast <\tau<\lambda_p $, the supremum

$ \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 $ u\in W_0^{1, 2}(\Omega) $ with $ \| \nabla u\|_{2}^2\leq4 \pi $. This is based on a method of energy estimate, which is developed by [14, 15, 16].