Nonlinear elliptic equation with measures as data and mixed or Fourier boundary conditions

1.1 Notations and Hypotheses In the sequel, Ω is a bounded domain in R (N ≥ 2), with a Lipschitz continuous boundary. n is the unit normal to ∂Ω outward to Ω. We denote by x · y the usual Euclidean product of two vectors (x, y) ∈ R × R ; the associated Euclidean norm is written |.|. The Lebesgue measure of a measurable subset E of R is denoted by |E|; σ is the Lebesgue measure on ∂Ω (i.e. the (N−1)-dimensional Hausdorff measure). For q ∈ [1,+∞], q denotes the conjugate exponent of q (i.e. 1/q + 1/q = 1). W (Ω) is the usual Sobolev space, endowed with the norm ||u||W 1,q(Ω) = ||u||Lq(Ω) + || |∇u| ||Lq(Ω). When Γ is a measurable subset of ∂Ω, W 1,q Γ (Ω) is the space of functions of W (Ω) which have a null trace on Γ. W 1− 1 q (∂Ω) denotes the Banach space of the traces on ∂Ω of functions of W (Ω), endowed with the norm ||f ||W 1−1/q,q(∂Ω) = inf { ||u||W 1,q(Ω) | u|∂Ω = f }