论文标题
半线性方程的正谐线解决方案
Positive harmonically bounded solutions for semi-linear equations
论文作者
论文摘要
对于某些空间$ x $的开放式集合$ u $，我们对$ u $上的半线性方程式的积极解决方案$ lu =φ（\ cdot，u）μ$感兴趣。这里$ l $可能是二阶（扩散过程的生成器）或截然不同的操作员（跳跃过程的生成器），$μ$是$ u $和$φ$的积极措施，是$ u $和$φ$是$ u \ u \ u \ u \ u \ u \ u \ mathbb {r} r} $ t（r），$ $ $ t是$ u $ cap { $ x \在U $中是连续的，在$ t = 0 $时增加和消失。 更确切地说，考虑到$ x $上的可测量功能$ h \ ge 0 $，是$ u $上的$ l $ harmonic，也就是说，$ u $上的$ lh = 0 $在$ u $上的持续真实，我们为存在的$ $ u = y $ $ $ $ $ $ $ $ $ u $ y $ u $ u $ $ $ u $ y $ $ $ u $ y $ u $ y $ u $ u $ u $ $ u $ $ $ （问题1）或$ u $上的$ u \ le H $，但$ u \ u \ not \ equiv 0 $ on $ u $（问题2）。 我们表明，这些问题等于存在某些积分方程的积极解决方案的问题$ u+kφ（\ cdot，u）= g $ on $ u $，$ k $是潜在的内核。我们在balayage空间的一般设置$（x，\ mathcal {w}）$中解决它们，从概率的角度来看，这对应于具有较强的Feller Resolvent的瞬态狩猎过程的设置。
For open sets $U$ in some space $X$, we are interested in positive solutions to semi-linear equations $ Lu=φ(\cdot,u)μ$ on $U$. Here $L$ may be an elliptic or parabolic operator of second order (generator of a diffusion process) or an integro-differential operator (generator of a jump process), $μ$ is a positive measure on $U$ and $φ$ is an arbitrary measurable real function on $U\times \mathbb{R}^+$ such that the functions $t\mapsto φ(x,t)$, $x\in U$, are continuous, increasing and vanish at $t=0$. More precisely, given a measurable function $h\ge 0$ on $X$ which is $L$-harmonic on $U$, that is, continuous real on $U$ with $Lh=0$ on $U$, we give necessary and sufficient conditions for the existence of positive solutions $u$ such that $u=h$ on $X\setminus U$ and $u$ has the same ``boundary behavior'' as $h$ on $U$ (Problem 1) or, alternatively, $u\le h$ on $U$, but $u\not\equiv 0$ on $U$ (Problem 2). We show that these problems are equivalent to problems of the existence of positive solutions to certain integral equations $u+Kφ(\cdot,u)=g$ on $U$, $K$ being a potential kernel. We solve them in the general setting of balayage spaces $(X,\mathcal{W})$ which, in probabilistic terms, corresponds to the setting of transient Hunt processes with strong Feller resolvent.
