Local estimates of the gradients of solution to a simplest regularisation for some class of nonuniformly elliptic

An estimate of $\max_{\Omega'}|u_x^\varepsilon|$, $\Omega'\subset\subset\Omega$, for solutions $u^\varepsilon$ to the family of equations
$$ -\frac d{dx_i}\,\frac{u_{x_i}}{\sqrt{1+u^2_x}}-\varepsilon\Delta u+a(x,u,u_x)=0,\qquad x\in\Omega,\quad\varepsilon\in(0,1], $$
with a non-differentiated lower term $a$ is given. A majorant in the estimate depends on $\max_{\Omega'}|u_x^\varepsilon|$ and the distance between $\Omega'$ and $\partial\Omega$, but does not depend on $\varepsilon$. The publication has relations with the work [2] and [3]. Bibliography: 4 titles.