Solvability of Elliptic Boundary Value Problems without Standard Landesman-Lazer Condition

In this paper we prove a very general result concerning solvability of the resonant problem: Δu + λku + g(x, u) = h(x); u = 0, x ∈ ∂Ω, which immediately gives three generalized Landesman-Lazer conditions. The most interesting application of the general result is concerned with the problem when λk = λ1, in which case we prove solvability results for it under conditions which are not the standard Landesman-Lazer condition or only partly enjoy it. Furthermore, we propose a new sign condition and give a comprehensive extension of a main result of Figueiredo and Ni.