Some existence's results for noncoercive “1-Laplacian” operator

Abstract

In this paper we study necessary and sufficient conditions on f and the first eigenfunctions for the 1-Laplacian, for equations of the form $\[\cases{-\mathrm{div}(\sigma)-\lambda =fu^{q-1},\quad u\geqslant 0\ \hbox{in}\ {\varOmega},\cr \noalign{\vskip3pt}\sigma\cdot \nabla u=|\nabla u|,\quad |\sigma|_{L^{\infty}({\varOmega})}\leqslant 1,\ u\in W_{0}^{1,1}({\varOmega})\cr}\]$ when q≤N/(N−1) and λ≥λ₁, λ₁ is the first eigenvalue for the 1-Laplacian.