(with $\sigma^T$ denoting its transpose) which satisfies $$\label{e.sigma}
\left| \sigma(e,x) \right| + \left| D_x \sigma(e,x) \right| + \left| D_\xi \sigma(e,x) \right| \leq C_0\qquad \mbox{in} \ \partial B_1\times {\mathbb{R}}^d.$$ For notational purposes, it is convenient to extend $\sigma(\cdot,x)$ (and $A(\cdot,x)$) to ${\mathbb{R}^d}\setminus\{0\}$ by defining $\sigma(\xi,x):= \sigma(\xi/|\xi|,x)$, which makes them $0$-homogeneous functions of their first argument. The Hamiltonian $$\label{e.H}
H \in C^{1}( {\mathbb{R}^d}\times{\mathbb{R}^d})$$ is assumed to satisfy, for every $t>0$ and $\xi,x\in{\mathbb{R}^d}$, $$\label{e.Hposhom}
H(t\xi,x) = t^pH(\xi,x)  \qquad \mbox{and} \qquad 
c_0\left|\xi\right|^p \leq H(\xi,x) \leq C_0\left|\xi\right|^p$$ and $$\label{e.HLip}
\left| D_x H( \xi,x) \right| + \left| \xi \right| \left| D_\xi H(\xi,x) \right| \leq C_0 \left| \xi \right|^p\qquad \mbox{in} \ \left({\mathbb{R}}^d\setminus\{0\}\right)\times {\mathbb{R}}^d.$$ The main structure condition on the coefficients is what we call the *Lions-Souganidis (LS) coercivity condition*, since it was introduced (albeit in a slightly different form) in [@LS0]: we suppose that, with the modulus $\rho$ fixed as above, $$\label{e.LScoercivity}
\inf\left\{ \mathcal C_{\sigma,H}(\xi,x) \,:\,  \xi,x\in{\mathbb{R}^d},  \ \left|\xi\right| \geq R  \right\} \geq \rho(R),$$ where the quantity $\mathcal C_{\sigma,H}$, which measures the coercivity of the equation, is defined by $$\begin{gathered}
 \label{}
\mathcal C_{\sigma,H}(\xi,x) := \inf_{\eta \in B_\kappa(\xi)} \bigg( \theta(1-2\theta)  \left( H(\eta,x)\right)^2 
-(1+\kappa)^3\left|\sigma(\eta,x)\right|^2\left|D_x\sigma(\eta,x)\right|^2|\xi|^2  \\
-\theta (1+\kappa)^2|\sigma(\eta,x)|^2 \left| \xi \right| \left( \left|D_xH(\eta,x)\right| + \kappa\left|D_\xi H(\eta,x)\right|\right) 
 \bigg).\end{gathered}$$ At first glance,  