u∈W1,2(Ω):−div(M(x,∇u))+a(x)u=−div(F). 0

where Ω is a bounded set in R^N and −div(M (x,∇u)) is a classical nonlinear differential operator, defined by a Carathéodory function M(x,ξ) satisfying, for some 0 < α ≤ β, and for almost every x ∈ Ω,

M(x,ξ)ξ≥α|ξ|² , |M(x,ξ)|≤β|ξ|, ∀ξ ∈ R^N ,

[M(x,ξ)−M(x,η)](ξ−η)>0, ∀ξ,η∈R^N , ξ̸=η.

Our key assumption is that the function 0 ≤a(x)∈L¹(Ω) and the vector-valued function F(x) are such that

∃R > 0 such that |F(x)|²≤Ra(x).

In addition, we prove the boundedness of u under the slightly stronger assumption that

∃R>0 and ∃p>2 such that |F(x)|^p≤Ra(x).

References