  We consider a class of stationary viscous Hamilton--Jacobi equations as $$
\left\{\begin{array}{l} \la u-{\rm div}(A(x) \nabla u)=H(x,\nabla u)\mbox{in
}\Omega, u=0{on}\partial\Omega\end{array} \right. $$ where $\la\geq 0$, $A(x)$
is a bounded and uniformly elliptic matrix and $H(x,\xi)$ is convex in $\xi$
and grows at most like $|\xi|^q+f(x)$, with $1 < q < 2$ and $f \in \elle {\frac
N{q'}}$. Under such growth conditions solutions are in general unbounded, and
there is not uniqueness of usual weak solutions. We prove that uniqueness holds
in the restricted class of solutions satisfying a suitable energy--type
estimate, i.e. $(1+|u|)^{\bar q-1} u\in \acca$, for a certain (optimal)
exponent $\bar q$. This completes the recent results in \cite{GMP}, where the
existence of at least one solution in this class has been proved.
