Abstract
We investigate the existence of solution for a linear elliptic problem posed in an unbounded cylinder where due to the presence of a nonpositive zero order term Lax-Milgram’s theorem does no apply. Since the domain is unbounded, Fredholm theory does not apply either. We are mainly interested in two types of solutions, solutions which tend to zero at infinity and solutions which are almost periodic in the longitudinal variable. This type of problems appears in the construction of correctors in wave homogenization taking into account the presence of boundary conditions.