论文标题
涉及一般有限和外部域上梯度的一些椭圆问题
Some elliptic problems involving the gradient on general bounded and exterior domains
论文作者
论文摘要
在本文中，我们考虑了有界域以及外部领域上的经典解决方案上存在积极的奇异解决方案。首先，我们考虑以下问题的积极奇异解决方案： \ begin {qore} \ label {eq_abst_1}-Δu=（1+g（x））| \ nabla u |^p \ qquad \ mbox {in} b_1，\ qquad u = 0 \ mbox {on} \; \; \ partial b_1，\ qquad \ mbox {and} \ end {equation} \ begin {qore} \ label {eq_abst_2}-ΔU= | \ nabla u |^p \ qquad \ mbox {in}ω，\ qquad u = 0 \ mbox {on} \; \; \部分ω。 \ end {equation} 在第一个问题中，$ b_1 $是$ \ mathbb {r}^n $中的单位球，第二美元$ω$是$ \ mathbb {r}^n $中的一个有限的平滑域。在这两种情况下，我们都假设$ n \ ge 3 $，$ \ frac {n} {n-1} <p <2 $，在第一个问题中，我们假设$ g \ ge 0 $是hölder连续功能，$ g（0）= 0 $。在这两种情况下，我们都会获得积极的单数解。 \\对于第二个方程式，我们还考虑了$ω$的外部域$ \ mathbb {r}^n $，其中$ n \ ge 3 $和$ p> \ frac {n} {n} {n-1} $。我们证明存在有限的正经典解决方案，并具有$ \ nabla u（x）\ cdot x> 0 $的附加属性，用于大$ | x | $。
In this article we consider the existence of positive singular solutions on bounded domains and also classical solutions on exterior domains. First we consider positive singular solutions of the following problems: \begin{equation} \label{eq_abst_1}-Δu = (1+g(x)) | \nabla u|^p \qquad \mbox{ in } B_1, \qquad u = 0 \mbox{ on } \;\; \partial B_1, \qquad \mbox{ and} \end{equation} \begin{equation} \label{eq_abst_2} -Δu = | \nabla u|^p \qquad \mbox{ in } Ω, \qquad u = 0 \mbox{ on } \;\; \partial Ω. \end{equation} In the first problem $B_1$ is the unit ball in $ \mathbb{R}^N$ and in the second $Ω$ is a bounded smooth domain in $ \mathbb{R}^N$. In both cases we assume $ N \ge 3$, $ \frac{N}{N-1}<p<2$ and in the first problem we assume $ g \ge 0$ is a Hölder continuous function with $g(0)=0$. We obtain positive singular solutions in both cases. \\ For the second equation we also consider the case of $Ω$ an exterior domain $ \mathbb{R}^N$ where $N \ge 3$ and $ p >\frac{N}{N-1}$. We prove the existence of a bounded positive classical solution with the additional property that $ \nabla u(x) \cdot x>0$ for large $|x|$.
