Existence of solutions for a resonant problem under landesman-lazer type conditions involving more general elliptic operators in divergence form

The present paper is concerned with the resonant problem −div (a (x, ∇u)) = λ1|u|p −2 u + f (x, u) − g(x) in Ω, where Ω is a bounded domain with smooth boundary in RN (N ≥ 2), p ∈ (1, ∞) and div(a (x, ∇u)) is a general elliptic operators in divergence form. By assuming a Landesman-Lazer type condition and using a variational method based on the Minimum Principle, we show the existence of a weak solution in the Sobolev space W0 1,p (Ω).