Existence results for some nonlinear and noncoercive anisotropic elliptic equations with Neumann boundary conditions

Existence results for some nonlinear .......

Résumé

Our aim in this work is to prove the existence of at least one renormalized solution for the anisotropic elliptic problem with Neumann boundary conditions

- div a(x,u,\nabla u) + H(x,u,\nabla u) + \alpha(x)|u|^{s-1}u = f         in  \Omega,
 a(x,u,\nabla u).\overrightarrow{n} = g                                                  on  \partial\Omega,

where   \Omega   is an open bounded subset of    IR^N    (N\geq 2),   and the data   f   belong to   L^{1}(\Omega)   and    g \in L^{1}(\partial \Omega).    a_{i}(x,r,\xi) and   H(x,r,\xi)    are two Carathéodory functions that verify some nonstandard conditions.