I am studying a stationary Hamilton-Jacobi-Bellman equation on $\mathbb{R}^N$ of the following type
$$-\Delta u+ H(\nabla u)+ c=f(x)$$
where the Hamiltonian can be assumed to be $H(q)=\frac{1}{\gamma}|q|^\gamma$ (or more in general is sufficiently regular). I am looking for a precise reference for the Holder reguarity of the solution $u$ (we assume that it exists) when $f\in C^{0,\alpha}$. Thank you!
in the case of bounded domains I can suggest the following reference Journal of differential equations vol 245 (2008) pages 2939–2957 . I hope this can help.