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

TL;DR

This paper proves the uniqueness of entire solutions to certain quasilinear p-Laplace type equations in \\mathbb{R}^n, including cases with measure data and sub-natural growth, under standard assumptions.

Contribution

It establishes the first known uniqueness results for solutions of quasilinear equations with measure data and nonlinear terms in the entire space.

Findings

Uniqueness of solutions for equations with measure data and nonlinearities.

Results apply to the p-Laplace operator and similar quasilinear operators.

Main theorems cover sub-natural growth case with specific measure conditions.

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.