On some nonlinear partial diffrential equations involving the 1-Laplacian

Let $\mathrm{\Omega }$ be a smooth bounded domain in ${\mathbb{R}}^N,N>1$ and let $n\in {\mathbb{N}}^{\ast }$. We are concerned here with the existence of nonnegative solutions $u\mathrm{\_}n$ in $BV(\mathrm{\Omega })$, to the problem $$(P\mathrm{\_}n)\{\begin{array}{l}-\mathrm{d}\mathrm{i}\mathrm{v}\sigma +2n(\int \mathrm{\_}\mathrm{\Omega }u-1){\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}^{+}(u)=0\text{in}\mathrm{\Omega },\sigma \cdot \mathrm{\nabla }u=\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }\text{in}\mathrm{\Omega },u\text{is not identically zero},-\sigma \cdot \overrightarrow{n}u=u\text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\overrightarrow{n}$ denotes the unit outer normal to $\mathrm{\partial }\mathrm{\Omega }$, and ${\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}^{+}(u)$ denotes some $L^{\mathrm{\infty }}(\mathrm{\Omega })$ function defined as: $${\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}^{+}(u)\mathrm{.}u=u^{+},0\le {\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}^{+}(u)\le 1.$$ Moreover, we prove the tight convergence of $u\mathrm{\_}n$ towards one of the first eingenfunctions for the first $1-$Laplacian Operator $-\mathrm{\Delta }\mathrm{\_}1$ on $\mathrm{\Omega }$ when $n$ goes to $+\mathrm{\infty }$.