  We prove the uniqueness property for a class of entire solutions to the
equation \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\,
\mathcal{A}(x,\nabla u) = \sigma, \quad u\geq 0 \quad \text{in } \mathbb{R}^n,
\\ \displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right.
\end{equation*} where $\sigma$ is a nonnegative locally finite measure in
$\mathbb{R}^n$, absolutely continuous with respect to the $p$-capacity, and
${\rm div}\, \mathcal{A}(x,\nabla u)$ is the $\mathcal{A}$-Laplace operator,
under standard growth and monotonicity assumptions of order $p$ ($1<p<\infty$)
on $\mathcal{A}(x, \xi)$ ($x, \xi \in \mathbb{R}^n$); the model case
$\mathcal{A}(x, \xi)=\xi | \xi |^{p-2}$ corresponds to the $p$-Laplace operator
$\Delta_p$ on $\mathbb{R}^n$.
  Our main results establish uniqueness of solutions to a similar problem,
\begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla
u) = \sigma u^q +\mu, \quad u\geq 0 \quad \text{in } \mathbb{R}^n, \\
\displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right.
\end{equation*}
  in the sub-natural growth case $0<q<p-1$, where $\mu, \sigma$ are nonnegative
locally finite measures in $\mathbb{R}^n$, absolutely continuous with respect
to the $p$-capacity, and $\mathcal{A}(x, \xi)$ satisfies an additional
homogeneity condition, which holds in particular for the $p$-Laplace operator.
