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arXiv 1704.04844math.AP
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测度框架下半椭圆椭圆型方程的各向异性奇性

Anisotropic singularities to semilienar elliptic equations in a measure framework

Huyuan Chen

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26 pages
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摘要：

本文研究椭圆型方程$$ -Δu+g(u)=2k\frac{\partial δ_0}{\partial x_N }+jδ_0\quad {\rm in}\quad\ \ B_1(0),\qquad u=0\quad {\rm on}\quad\ \ \partial B_1(0), $$的很弱解，其中$k>0$，$j\ge0$，$B_1(0)$表示$\mathbb{R}^N$中以原点为中心的单位球，$N\geq2$，$g:\mathbb{R}\to\mathbb{R}$是奇的、非减的$C^1$函数，$δ_0$是集中在原点的狄拉克质量，$\frac{\partialδ_0}{\partial x_N}$定义在分布意义上，即$$ \langle\frac{\partial δ_0}{\partial x_N},ζ\rangle=\frac{\partialζ(0)}{\partial x_N} , \qquad \forall ζ\in C^1_0(B_1(0)). $$。在积分次临界假设下弱解$u_{k,j}$的存在性$$\int_1^{\infty}g(s)s^{-1-\frac{N+1}{N-1}}ds<+\infty.$$进一步证明了$u_{k,j}$在原点具有各向异性奇异性，并将$\{u_{k,0}\}_k$的奇性$u_{k,0}$和极限看作是$k\to\infty$ .我们提出的约束非线性$g(u)$，我们只需要在主值意义上的可积性，由于奇点只在原点。这使得我们能够在更大的非线性范围内搜索非常弱的解。

英文摘要：

The purpose of this article is to study very weak solutions of elliptic equation $$ -Δu+g(u)=2k\frac{\partial δ_0}{\partial x_N }+jδ_0\quad {\rm in}\quad\ \ B_1(0),\qquad u=0\quad {\rm on}\quad\ \ \partial B_1(0), $$ where $k>0$, $j\ge0$, $B_1(0)$ denotes the unit ball centered at the origin in $\mathbb{R}^N$ with $N\geq2$, $g:\mathbb{R}\to\mathbb{R}$ is an odd, nondecreasing and $C^1$ function, $δ_0$ is the Dirac mass concentrated at the origin and $\frac{\partialδ_0}{\partial x_N}$ is defined in the distribution sense that $$ \langle\frac{\partial δ_0}{\partial x_N},ζ\rangle=\frac{\partialζ(0)}{\partial x_N} , \qquad \forall ζ\in C^1_0(B_1(0)). $$ We obtain that the above problem admits a unique very weak solution $u_{k,j}$ under the integral subcritical assumption $$\int_1^{\infty}g(s)s^{-1-\frac{N+1}{N-1}}ds<+\infty.$$ Furthermore, we prove that $u_{k,j}$ has anisotropic singularity at the origin and we consider the odd property $u_{k,0}$ and limit of $\{u_{k,0}\}_k$ as $k\to\infty$. We pose the constraint on nonlienarity $g(u)$ that we only require integrability in the principle value sense, due to the singularities only at the origin. This makes us able to search the very weak solutions in a larger scope of the nonlinearity.

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