We consider the solution $u_p$ to the Neumann problem for the $p$--Laplacian equation with the normal component of the flux across the boundary given by $g\in L^\infty(\partial\Omega)$. We study the behaviour of $u_p$ as $p$ goes to $1$ showing that they converge to a measurable function $u$ and the gradients $|\nabla u_p|^{p-2}\nabla u_p$ converge to a vector field $z$. We prove that $z$ is bounded and that the properties of $u$ depend on the size of $g$ measured in a suitable norm: if $g$ is small enough, then $u$ is a function of bounded variation (it vanishes on the whole domain, when $g$ is very small) while if $g$ is large enough, then $u$ takes the value $\infty$ on a set of positive measure. We also prove that in the first case, $u$ is a solution to a limit problem that involves the $1-$Laplacian. Finally, explicit examples are shown.

Non-linear elliptic Neumann problems, possibly in irregular domains and with data affected by low integrability properties, are taken into account. Existence, uniqueness and continuous dependence on the data of generalized solutions are established under a suitable balance between the integrability of the datum and the (ir)regularity of the domain. The latter is described in terms of isocapacitary inequalities. Applications to various classes of domains are also presented. Résumé: Nous considérons des problèmes de Neumann pour des équations elliptiques non linéaires dans des domaines éventuellement non réguliers et avec des données peu régulières. Un équilibre entre l'intégrabilité de la donnée et l'(ir)régularité du domaine nous permet d'obtenir l'existence, l'unicité et la dépendance continue de solutions généralisées. L'irrégularité du domaine est décrite par des inégalités « isocapacitaires ». Nous donnons aussi des applications à certaines classes de domaines.

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is where Ω is a bounded open subset of , N > 2, 2-1/N < p < N , a belongs to L ∞(Ω), , f is a function in L 1(Ω), b is a function in and 0 ≤ λ < λ *(N,p,r), for some r and λ *(N,p,r).