Uniqueness of entire solutions to quasilinear equations of $p$-Laplace type

We prove the uniqueness property for a class of entire solutions to the equation

$$\{\begin{array}{l}-\mathrm{d}\mathrm{i}\mathrm{v}\mathcal{A}(x,\mathrm{\nabla }u)=\sigma ,u\ge 0\text{in}{\mathbb{R}}^n,\\ \underset{|x|\to \mathrm{\infty }}{lim inf}u=0,\end{array}$$

where $\sigma$ is a nonnegative locally finite measure in ${\mathbb{R}}^n$, absolutely continuous with respect to the $p$-capacity, and $\mathrm{d}\mathrm{i}\mathrm{v}\mathcal{A}(x,\mathrm{\nabla }u)$ is the $\mathcal{A}$-Laplace operator, under standard growth and monotonicity assumptions of order $p$ ( $1<p<\mathrm{\infty }$) 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 ${\mathrm{\Delta }}_p$ on ${\mathbb{R}}^n$. Our main results establish uniqueness of solutions to a similar problem,

$$\{\begin{array}{l}-\mathrm{d}\mathrm{i}\mathrm{v}\mathcal{A}(x,\mathrm{\nabla }u)=\sigma u^q+\mu ,u\ge 0\text{in}{\mathbb{R}}^n,\\ \underset{|x|\to \mathrm{\infty }}{lim inf}u=0,\end{array}$$

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.