Nguyen Cong Phuc

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.

Comparison estimates are an important technical device in the study of regularity problems for quasilinear possibly degenerate elliptic and parabolic equations. Such tools have been employed indispensably in many papers of Mingione, Duzaar-Mingione, and Kuusi-Mingione, etc. on certain measure datum problems to obtain pointwise bounds for solutions and their full or fractional derivatives in terms of appropriate linear or nonlinear potentials. However, a comparison estimate for $p$-Laplace type elliptic equations with measure data is still unavailable in the strongly singular case $1< p\leq \frac{3n-2}{2n-1}$, where $n\geq 2$ is the dimension of the ambient space. This issue will be completely resolved in this work by proving a comparison estimate in a slightly larger range $1<p<3/2$. Applications include a `sublinear' Poincar\'e type inequality, pointwise bounds for solutions and their derivatives by Wolff's and Riesz's potentials, respectively. Some global pointwise and weighted estimates are also obtained for bounded domains, which enable us to treat a quasilinear Riccati type equation with possibly sublinear growth in the gradient.

We prove the uniqueness property for a class of entire solutions to the equation \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $\end{document} 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 $. Our main results establish uniqueness of solutions to a similar problem, \begin{document}$ \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, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $\end{document} 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.