论文标题
全球方程式的积极解决方案的全局估计值
Global pointwise estimates of positive solutions to sublinear equations
论文作者
论文摘要
我们将阳性解决方案的双边估算值$ u $ $ $ $ $ \ [U = \ MathBf {G}（σU^Q） + F \ Quad \ textrm {in} \，\，\，ω，\]，对于$ 0 <q <1 $，其中$σ\ ge 0 $是可衡量的函数，是可衡量的函数，或radon量，$ f \ ge 0 $，$ f \ ge $ \ nisbf iS $ \ g}在$ω\timesΩ$上。我们的主要结果包括存在的存在标准和解决方案的唯一性，可用于准 - 列表或准里率可修改的内核$ g $。 结果，我们获得了双边估计以及存在和独特性，用于正面解决方案$ u $，可能是无限的，用于涉及分数laplacian的椭圆形方程， \ [（-Δ）^{\fracα{2}} u =σu^q +μ\ quad \ quad \ textrm {in} \，\，\，\，\ qquad u = 0 \，\，\，\，\，\ \ \ \ \ \ \ \ \ textrm {in}在有限的统一域$ 0 <α\ le 2 $上或整个空间$ \ mathbf {r}^n $，a球或半空间，以$ 0 <α<n $ 0 <α<n $。
We give bilateral pointwise estimates for positive solutions $u$ to the sublinear integral equation \[ u = \mathbf{G}(σu^q) + f \quad \textrm{in} \,\, Ω,\] for $0 < q < 1$, where $σ\ge 0$ is a measurable function, or a Radon measure, $f \ge 0$, and $\mathbf{G}$ is the integral operator associated with a positive kernel $G$ on $Ω\timesΩ$. Our main results, which include the existence criteria and uniqueness of solutions, hold for quasi-metric, or quasi-metrically modifiable kernels $G$. As a consequence, we obtain bilateral estimates, along with the existence and uniqueness, for positive solutions $u$, possibly unbounded, to sublinear elliptic equations involving the fractional Laplacian, \[ (-Δ)^{\fracα{2}} u = σu^q + μ\quad \textrm{in} \,\, Ω, \qquad u=0 \, \, \textrm{in} \,\, Ω^c, \] where $0<q<1$, and $μ, σ\ge 0$ are measurable functions, or Radon measures, on a bounded uniform domain $Ω\subset \mathbf{R}^n$ for $0 < α\le 2$, or on the entire space $\mathbf{R}^n$, a ball or half-space, for $0 < α<n$.
