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.
Categories: math.AP