论文标题
半椭圆方程的非负解的估计值
Estimates of Nonnegative Solutions to Semilinear Elliptic Equations
论文作者
论文摘要
令$ l $为域中的第二阶均匀椭圆形差异操作员$ d $ of $ \ mathbb {r}^{d} $，$ψ：\ Mathbb {r} _+\ to \ to \ to \ mathbb {r} _ r} _局部界限的Borel可测量功能。在适当的条件下，我们确定一个函数$φ$，其值$] 0,1] $，以便每种不平等的解决方案$ -lu+-lu+ξψ（u）\ geq g $ in $ d $中 $$ u（x）\ geq p（x）\，φ\ left（\ frac {g_d（ξψ（p））（x）}} {p（x）} \ right） $$ 其中$ p = g_dg $是$ g $的绿色功能。功能$φ$完全由$ψ$确定，不取决于$ L，D，ξ$或$ G $。
Let $L$ be a second order uniformly elliptic differential operator in a domain $D$ of $\mathbb{R}^{d}$, $ψ:\mathbb{R}_+\to \mathbb{R}_+$ be a nondecreasing continuous function and let $ξ,g:D\to\mathbb{R}_+$ be locally bounded Borel measurable functions. Under appropriate conditions, we determine a function $φ$ with values in $]0,1]$ such that for every nonnegative solution to inequality $-Lu+ξψ(u) \geq g$ in $D$ and for every $x\in D$, $$ u(x)\geq p(x)\,φ\left(\frac{G_D(ξψ(p))(x)}{p(x)}\right) $$ where $p=G_Dg$ is the Green function of $g$. The function $φ$ is completely determined by $ψ$ and does not depend on $L,D,ξ$ or $g$.
