Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations

We consider a class of stationary viscous Hamilton-Jacobi equations as lambda u - div(A(x)del u) = H(x, del u) in Omega u = 0 on partial derivative Omega where lambda >= 0, A(x) is a bounded and uniformly elliptic matrix and H(x, xi) is convex in xi and grows at most like \xi\(q) + f (x), with 1 < q < 2 and f is an element of L-N/q'(Omega). Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy-type estimate, i.e. (1 + \u\)((q) over bar -1) u is an element of H-0(1) (Omega), for a certain (optimal) exponent (q) over bar. This completes the recent results in [15], where the existence of at least one solution in this class has been proved.