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
L
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).