Neumann problems for nonlinear elliptic equations with lower order terms

In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is λ | u | p − 2 u − Δ p u − div ( c ( x ) | u | p − 2 u ) + b ( x ) | ∇ u | p − 2 ∇ u = f in Ω , | ∇ u | p − 2 ∇ u + c ( x ) | u | p − 2 u ⋅ n ̲ = 0 on ∂ Ω where Ω is a bounded domain of R N , N ≥ 2 , with Lipschitz boundary, 1 0 , the datum f belongs to the dual space of W 1 , p ( Ω ) or to Lebesgue space L 1 ( Ω ) . Finally the coefficients b , c belong to appropriate Lebesgue spaces or Lorentz spaces. Existence results for weak solutions or renormalized solutions are proved under smallness assumptions on the coefficients b and c .