Existence of solutions of certain quasilinear elliptic equations in $\mathbb R^N$ without conditions at infinity

The paper deals with conditions for the existence of solutions of the equations
$$ -\sum_{i=1}^nD_iA_i(x,u,Du)+A_0(x,u)=f(x),\quad x\in\mathbb R^n, $$
considered in the whole space $\mathbb R^n$, $n\ge2$. The functions $A_i(x,u,\xi)$, $i=1,\dots,n$, $A_0(x,u)$, and $f(x)$ can arbitrarily grow as $|x|\to\infty$. These functions satisfy generalized conditions of the monotone operator theory in the arguments $u\in\mathbb R$ and $\xi\in\mathbb R^n$. We prove the existence theorem for a solution $u\in W_{\mathrm{loc}}^{1,p}(\mathbb R^n)$ under the condition $p>n$.