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In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class ofnon coercive nonlinear problems whose prototype is
$$\left\{- \hbox{div}( a(x)(1+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u) +b(x)(1+|\nabla u|^{2})^{\frac{\lambda}{2}} =f\hbox{in}\quad\Omega, u=0\hbox{on} \quad \partial\Omega,\right.$$ 
where Ω is a bounded open subset of ${\mathbb{R}}^N$ 
, N > 2, 2-1/N < p < N , a belongs to L ∞(Ω), $a(x)\ge\alpha_0>0$ 
,f is a function inL 1(Ω), b is a function in $L^r(\Omega)$ 
and 0 ≤ λ < λ *(N,p,r), for some r and λ *(N,p,r).
