论文标题
P-Laplace类型的准线性方程的整个解决方案的唯一性
Uniqueness of entire solutions to quasilinear equations of p-Laplace type
论文作者
论文摘要
我们证明了一类方程\ begin {qore*} \ left \ {\ oken {array} {ll} {ll} - {\ rm div} \，\ nathcal {a}（x，x，x，\ nabla u \ qeq q q eq q qeq 0 \ Mathbb {r}^n，\\ \ displayStyle {\ liminf_ {| x | \ rightArrow \ infty \ infty}}} \，u = 0，\ end \ end {array} \ right。 \ end {equation*}其中$σ$是$ \ mathbb {r}^n $中的非负局部有限量度$ \ MATHCAL {a}（x，ξ）$（$ x，ξ\ in \ Mathbb {r}^n $）上的订单$ p $（$ 1 <p <\ infty $）的单调假设;模型情况$ \ MATHCAL {A}（x，ξ）=ξ| ξ|^{p-2} $对应于$ p $ -laplace Operator $Δ_p$上的$ \ m \ mathbb {r}^n $。 我们的主要结果确定了解决类似问题的唯一性，\ begin {equination*} \ left \ {\ begin {array} {ll} {ll} - {\ rm div} \，\ m athcal {A} \ Mathbb {r}^n，\\ \ displayStyle {\ liminf_ {| x | \ rightArrow \ infty \ infty}}} \，u = 0，\ end \ end {array} \ right。 \ end {equation*} in the sub-natural growth case $0<q<p-1$, where $μ, σ$ are nonnegative locally finite measures in $\mathbb{R}^n$, absolutely continuous with respect to the $p$-capacity, and $\mathcal{A}(x, ξ)$ satisfies an additional homogeneity condition, which holds in particular for the $p$-Laplace operator.
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) = σ, \quad u\geq 0 \quad \text{in } \mathbb{R}^n, \\ \displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} where $σ$ 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 $\mathcal{A}(x, ξ)$ ($x, ξ\in \mathbb{R}^n$); the model case $\mathcal{A}(x, ξ)=ξ| ξ|^{p-2}$ corresponds to the $p$-Laplace operator $Δ_p$ 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) = σu^q +μ, \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 $μ, σ$ are nonnegative locally finite measures in $\mathbb{R}^n$, absolutely continuous with respect to the $p$-capacity, and $\mathcal{A}(x, ξ)$ satisfies an additional homogeneity condition, which holds in particular for the $p$-Laplace operator.
