Normalized solutions to 1-Laplace equations with critical growth term

We study normalized solutions to the 1-Laplace equation $$-div\left(\frac{Du}{\mathrm{\mid }Du\mathrm{\mid }}\right)+\lambda \frac{u}{\mathrm{\mid }u\mathrm{\mid }}=f(u)+\mathrm{\mid }u{\mathrm{\mid }}^{1^{\ast }-2}u,\text{quadin}{\mathbb{R}}^N$$ subject to the constraint ${\int }_{{\mathbb{R}}^N}\mathrm{\mid }u\mathrm{\mid }dx=m$, where $N\ge 2$, $1^{\ast }=\frac{N}{N-1}$ is the critical Sobolev exponent for the embedding from BV into $L^p$, $\lambda ,m\in {\mathbb{R}}^{+}$ and $f:\mathbb{R}\to \mathbb{R}$ is a continuous nonlinear perturbation. We first establish a Pohozaev-type identity for the equation and use it to study the associated critical energy, and then prove the existence of normalized solutions via nonsmooth critical point theory. A key feature of our approach is the proof of the existence of a Palais-Smale sequence for a Lipschitz continuous functional under a Lipschitz continuous constraint. This is significantly different in methodology from previous problems with $C^2$ constraint.