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: Delta u + lambda(k) u + g(x, u) = h(x) u = 0, x is an element of partial derivative Omega, which immediately gives three generalized Landesman-Lazer conditions. The most interesting application of the general result is concerned with the problem when lambda(k) = lambda(1), 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.