Neumann problems for nonlinear elliptic equations with L1 data

Abstract

In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype iswhen f is just a summable function. Our approach allows also to deduce a stability result for renormalized solutions and an existence result for operator with a zero order term. Mathematics Subject Classification:MSC 2000 : 35J25

Gradient estimates for problems with Orlicz growth

“…Corresponding results for systems are provided in [40]. Let us refer also to some other attempts [5,7,21,12,23,28,30,43,44,46].…”

Uniqueness for Neumann problems for nonlinear elliptic equations

“…for almost every x ∈ Ω and for every ξ ∈ R N . Moreover a is strongly monotone, that is a constant β > 0 exists such that 8) for almost every x ∈ Ω and for every ξ, η ∈ R N , ξ = η. We assume that Φ : Ω × R → R N is a Carathéodory function which satisfies the following "growth condition"…”

“…The existence for solutions having null median to problem (1.1) when c = 0 are proved in [8]. We explicitly remark that when the datum f has a lower summability, i.e.…”

“…The main novelty of this article is to prove uniqueness (up to additive constants) results for renormalized solutions to problem (1.1) having null median and whose existence has been proved in [8].…”

“…When c(x) = 0 and f is an element of the dual space of the Sobolev space W 1,p (Ω), the existence and uniqueness (up to additive constants) of weak solutions to problem (1.1) is consequence of the classical theory of pseudo-monotone operators (cfr. [25,26]), while existence results for weak solutions to problem (1.1) when the lower order term appears have been proved in [8].…”