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

Abstract

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.