A Sufficient Condition for a Blow-Up in the Space of Absolutely Continuous Functions for the Very Weak Solution

Given an open bounded smooth set $\Omega$ in $I{R}^N,N⩾3$, we provide a sufficient condition on the data $f$ integrable with respect to the distance $\delta$, to ensure the blow-up of the gradient of the very weak solution for the Dirichlet equation.