We consider the following problem:

\begin{alignat*}{2} -\text{div}(A(x,u)\nabla u)&=u^s+f(x) & \quad &\text{in }\varOmega, \\ u(x)&\ge0 & & \text{in }\varOmega, \\ u(x)&=0 & & \text{on }\p\varOmega, \end{alignat*}

where $\varOmega$ is an open bounded subset of $\mathbb{R}^N$, $N\ge3$, and

$$ A:\varOmega\times\mathbb{R}\rightarrow M_{N\times N} $$

is an elliptic matrix such that when $u\to\infty$ is non-coercive.

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
$$ \left\{ - \hbox{div}( a(x)(1+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u) +b(x)(1+|\nabla u|^{2})^{\frac{\lambda}{2}} =f \hbox{in} \quad \Omega, u=0 \hbox{on} \quad \partial\Omega, \right. $$
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 in L1(Ω), b is a function in $L^r(\Omega)$ and 0 ≤ λ < λ *(N,p,r), for some r and λ *(N,p,r).