Classical global solutions for a class of Hamilton-Jacobi equations

Let V be a real-valued function of class C2 on ℝn, n ≥ 2, which vanishes if {pipe}x{pipe} ≤ R and, for some ∈ > 0, satisfies ∂αx V(x) = O({pipe}x{pipe}-∈-{pipe}α{pipe}), as {pipe}x{pipe} → ∞, for {pipe}α{pipe} ≤ 2. Using a global inverse function theorem of Hadamard, we showthat if R is sufficiently large, then the Hamilton-Jacobi equation of eikonal type {pipe}∇u{pipe}2+V(x) = k2, with k > 0, has a C1 solution on ℝn \ {0}. © Birkhäuser/Springer Basel AG 2010.