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

We study the boundary-value problem{F(D2u,Du,u,x)+λu=f(x,u)in Ω,u=0on ∂Ω, where the second order differential operator F is of Hamilton–Jacobi–Bellman type, f is sub-linear in u   at infinity and Ω⊂RNΩ⊂RN is a regular bounded domain. We extend the well-known Landesman–Lazer conditions to study various bifurcation phenomena taking place near the two principal eigenvalues associated to the differential operator. We provide conditions under which the solution branches extend globally along the eigenvalue gap. We also present examples illustrating the results and hypotheses.