In this article, we propose a new non-linear stabilisation approach based on the popular linear parameter-varying control techniques. The regional state-feedback control problem of polynomial non-linear systems will be studied using rational Lyapunov functions of states. By bounding the variation rates of each state, the domain of attraction will be embedded in the region specified by the non-linear vector field. As a result, the state-feedback stabilisation conditions will be formulated as a set of polynomial matrix inequalities and can be solved efficiently by sum-of-squares programming. The resulting Lyapunov matrix and state-feedback gains are typically state-dependent rational matrix functions. This approach is also extended to a class of output-dependent non-linear systems where the stabilising output-feedback controller can be synthesised using rational Lyapunov functions of outputs. Finally, several examples will be used to demonstrate the proposed stabilisation approach and clarify the effect of various choices of Lyapunov function forms and state constraints.