Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations

Abstract

We consider a class of stationary viscous Hamilton-Jacobi equations as {λu - div(A(x)▽u) = H(x, ▽u) in Ω, {u = 0 on ∂Ω where λ ≥ 0, A(x) is a bounded and uniformly elliptic matrix and H(x, ξ) is convex in ξ and grows at most like ξ q + f(x), with 1 < q < 2 and f ∈ LN/q' (Ω). 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 )q̄-1 u ∈ H01 (Ω), for a certain (optimal) exponent q̄. This completes the recent results in [15], where the existence of at least one solution in this class has been proved.