and we made no attempt to optimize our proof to obtain the best exponent. It is less obvious that the limiting Hölder exponent of $\tfrac 27$ for the regularity of $\overline H$ can be improved, at least in the general quasilinear setting (it is easy to show that $\overline H$ is Lipschitz in the semilinear case). Note that in the statement of the theorem we used the notation $C^{0,\beta-}_{\mathrm{loc}}({\mathbb{R}^d}) := \cap_{0<\gamma<\beta} C^{0,\gamma}_{\mathrm{loc}}({\mathbb{R}^d})$.

Examples {#s.assumptions}
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In this subsection, we check that the motivating examples  and  satisfy the (LS) condition .

The assumption  is redundant in the case $p>1$, that is, we may remove the (LS) condition in the case of a superlinear Hamiltonian. Therefore our results apply to a large family of quasilinear, viscous and, in general, nonconvex Hamilton-Jacobi equations.

To check this, it is enough to show that, if $\mathcal C_{\sigma,H}(\xi,x)\leq  K$ for some $K\geq 1$, then $|\xi|\leq R_K$ for a sufficiently large real number $R_K$. The condition $\mathcal C_{\sigma,H}(\xi,x) \leq K$ implies that there exists $\eta\in B_\kappa(\xi)$ such that $$\begin{gathered}
K \geq \theta(1-2\theta) c_0^2|\eta|^{2p} -\theta (1+\kappa)^2|\sigma|^2C_0 (1+|\eta|^p) |\xi|
 -(1+\kappa)^3|\sigma|^2|D_x\sigma|^2|\xi|^2 \\ -\theta \kappa (1+\kappa)^2 |\sigma|^2C_0 (1+|\eta|^{p-1})|\xi|, \end{gathered}$$ where $\sigma$ and $D_x\sigma$ are evaluated at $(\eta,x)$. If we choose $\theta=\frac14$, 