On a generalization of the Landesman-Lazer condition and Neumann problem for nonuniformly semilinear elliptic equations in an unbounded domain with nonlinear boundary condition

This paper deals with the existence of weak solutions of Neumann problem for a nonuniformly semilinear elliptic equation: (Formula presented.) where Ω ⊂ RN, N ≥ 3 is an unbounded domain with smooth and bounded boundary ∂Ω, Ω¯ = Ω ∪ ∂Ω, h(x) ∈ (Formula presented.) (Ω¯), a(x) ∈ C(Ω¯), a(x) → + ∞ as |x| → + ∞, f(x, s), x ∈ Ω, g(x, s), x ∈ ∂Ω are Carathéodory, k(x) ∈ L2 (Ω), Θ(x) ∈ L∞ (Ω¯), Θ(x) ≥ 0. Our arguments is based on the minimum principle and rely essentially on a generalization of the Landesman-Lazer type condition.