Is there anyone checked on the paper Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks by A. Fiorenza et al. carefully?
In part 3.2 Proof of Theorem 1.2 of it, I wonder what if the $L^p$ norm of the derivative of $u$ is infinity at the first place, since if so, the last step of the argument seems to be invalid...
