In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is
$$\left\{-\text{div}\left(a\left(x\right)\right(1+{\left|\nabla u\right|}^2{)}^{\frac{p-2}{2}}{\nabla u)+b\left(x\right)(1+\left|\nabla u\right|}^2{)}^{\frac{\lambda }{2}}=f\text{in}\Omega ,u=0\text{on}\partial \Omega ,\right.$$
where Ω is a bounded open subset of ${ℝ}^N$, N > 2, 2-1/, belongs to  (Ω), $a\left(x\right)\ge {\alpha }_0>0$, is a function in (Ω), is a function in $L^r\left(\Omega \right)$ and 0 ≤ λ < λ *(), for some and λ *().