We present a non-compact 4 + 1 dimensional model with a local strong four-fermion interaction supplementing it with gravity. In the strong coupling regime it reveals the spontaneous translational symmetry breaking which eventually leads to the formation of domain walls, or thick 3-branes, embedded in the AdS₅ manifold. To describe this phenomenon we construct the appropriate low-energy effective Action and find kink-like vacuum solutions in the quasi-flat riemannian metric. We discuss the generation of ultra-low-energy 3 + 1 dimensional physics and we establish the relation among the bulk five dimensional gravitational constant, the brane Newton's constants and the curvature of AdS₅ space-time. The plausible relation between the compositeness scale of the scalar matter and the symmetry breaking scale is shown to support the essential decoupling of branons, the scalar fluctuations of the brane, from the Standard Model matter, supporting their possible role in the dark matter saturation. The induced cosmological constant on the brane does vanish due to exact cancellation of matter and gravity contributions.