On some nonlinear partial diffrential equations involving the 1-Laplacian

Abstract

Let $\Omega $ be a smooth bounded domain in $\R^N, N>1$ and let $n\in \N^*$. We are concerned here with the existence of nonnegative solutions $u\_n$ in $BV(\Omega)$, to the problem $$(P\_n) \begin{cases} -{\rm div} \sigma +2n (\int\_ \Omega u -1) {\rm sign}^+ (u)=0 \quad \text{in} \Omega, \sigma \cdot \nabla u= |\nabla u| \quad \text{in} \Omega, u \text{\rm is not identically zero}, -\sigma \cdot \overrightarrow {n} u=u \quad \text{on} \partial\Omega, \end{cases}$$ where $\overrightarrow {n}$ denotes the unit outer normal to $\partial\Omega$, and ${\rm sign}^+(u)$ denotes some $L^{\infty}(\Omega)$ function defined as: $${\rm sign}^+ (u). u =u^+, 0 \leq {\rm sign}^+(u) \leq 1.$$ Moreover, we prove the tight convergence of $u\_n$ towards one of the first eingenfunctions for the first $1-$Laplacian Operator $-\Delta\_1$ on $\Omega$ when $n$ goes to $+\infty$.