This paper establishes isomorphism for the Laplace operator in weighted Sobolev spaces (WSS). These spaces are similar to standard Sobolev spaces, but they are endowed with weights prescribing functions' growth or decay at infinity. Although well established in the whole space, these weighted results do not apply in the specific hypothesis of periodicity. This kind of problem appears studying singularity perturbed domains (roughness, sieves, porous media, etc). When zooming on a single pertubation pattern, one often ends with a periodic problem set on an infinite strip. We present a unified framework that enables a systematic treatment of such problems. We provide existence and uniqueness of solutions in our WSS. This gives a refined description of solutions behavior at infinity which is of importance in the multi-scale context. These isomorphism results hold for any weight exponent and any regularity index m. We then identify these solutions with the convolution of a Green function (specific to periodical infinite strips) and the given data.