Landesman-Lazer type results for second order Hamilton-Jacobi-Bellman equations

where the second order differential operator F is of Hamilton-Jacobi-Bellman (HJB) type, that is, F is a supremum of linear elliptic operators, f is sublinear in u at infinity, and Ω ⊂ R is a regular bounded domain. HJB operators have been the object of intensive study during the last thirty years – for a general review of their theory and applications we refer to [26], [33], [43], [14]. Well-known examples include the Fucik operator ∆u+ bu + au− ([27]), the Barenblatt operator max{a∆u, b∆u} ([10], [31]), and the Pucci operator M+λ,Λ(Du) ([37], [15]). To introduce the problem we are interested in, let us first recall some classical results in the case when F is the Laplacian and λ ∈ (−∞, λ2), (we shall denote with λi the i-th eigenvalue of the Laplacian). If f is independent of u the solvability of (1.1) is a consequence of the Fredholm alternative, namely, if λ 6= λ1, problem (1.1) has a solution for each f , while if λ = λ1 (resonance) it has solutions if and only if f is orthogonal to φ1, the first eigenfunction of the Laplacian. The existence result in the non-resonant case extends to nonlinearities f(x, u) which grow sub-linearly in u at infinity, thanks to Krasnoselski-Leray-Schauder degree and fixed point theory, see [1]. A fundamental result, obtained by Landesman and Lazer [34] (see also [30]), states that in the resonance case λ = λ1 the problem