论文标题
用于椭圆形半线性和准线性方程应用的均质性问题的非线性潜在估计值
Nonlinear potential estimates for sublinear problems with applications to elliptic semilinear and quasilinear equations
论文作者
论文摘要
我们对非线性潜在估计值及其应用的调查最近获得了额定性问题的阳性解决方案 \ [ u = \ mathbf {g}（σu^q） + f \ quad \ textrm {in} \，\，ω， \]其中$ 0 <q <1 $，$σ\ ge 0 $是$ω$中的ra度量，$ f \ ge 0 $是一个可测量的函数，$ \ mathbf {g} $是一个线性积分运算符，带有正核$ g $ on $ω\ times timpes $。对于准字体（或准里率修改）内核$ g $，这些双边估计值产生存在标准和解决方案的唯一性$ u \ in l^q _ {\ rm loc}（Ω，ω，σ）$。 应用被视为涉及（分数）拉普拉斯的半线性椭圆方程 \ [ （-Δ）^{\fracα{2}} u =σu^q +μ\ quad \ textrm {in} \，\，\，ω，\ qquad u = 0 \，\，\，\，\，\ textrm {in} \ \]这里$ 0 <q <1 $，$μ，σ\ ge 0 $是ra的措施，$ω$是$ {\ mathbb r}^n $中的一个有限的统一域，如果$ 0 <α\ le 2 $或整个空间$ {\ Mathbb r}^n $，则是$ 0 <$ 0 <n $ 0 <n， 这些结果的类似物是针对涉及整个空间上$ p $ laplace运算符$ {\ mathbb r}^n $， \ [-Δ_Pu =σu^q +μ\ Quad \ textrm {in} \，\，\，{\ Mathbb r}^n，\ qquad \ liminf_ {x \ to \ to \ infty} \] 其中$ 0 <q <p-1 $，$μ，σ\ ge 0 $是ra措施。具有$ \ MATHCAL {a} $的更通用的quasilinear方程 - 拉普拉斯运算符$ {\ rm div} \ mathcal {a}（x，x，\ nabla u）$也涵盖了$Δ_P$。
We give a survey of nonlinear potential estimates and their applications obtained recently for positive solutions to sublinear problems of the type \[ u = \mathbf{G}(σu^q) + f \quad \textrm{in} \,\, Ω, \] where $0 < q < 1$, $σ\ge 0$ is a Radon measure in $Ω$, $ f \ge 0$ is a measurable function, and $\mathbf{G}$ is a linear integral operator with positive kernel $G$ on $Ω\timesΩ$. For quasi-metric (or quasi-metrically modifiable) kernels $G$, these bilateral pointwise estimates yield existence criteria and uniqueness of solutions $u \in L^q_{\rm loc} (Ω, σ)$. Applications are considered to semilinear elliptic equations involving the (fractional) Laplacian, \[ (-Δ)^{\fracα{2}} u = σu^q + μ\quad \textrm{in} \,\, Ω, \qquad u=0 \, \, \textrm{in} \,\, Ω^c. \] Here $0<q<1$, $μ, σ\ge 0$ are Radon measures, and $Ω$ is a bounded uniform domain in ${\mathbb R}^n$, if $0 < α\le 2$, or the entire space ${\mathbb R}^n$, a ball or half-space, if $0 < α<n$. Analogues of these results are presented for elliptic equations involving the $p$-Laplace operator on the entire space ${\mathbb R}^n$, \[ -Δ_p u = σu^q + μ\quad \textrm{in} \,\, {\mathbb R}^n, \qquad \liminf_{x\to \infty} u(x)=0, \] where $0<q<p-1$, and $μ, σ\ge 0$ are Radon measures. More general quasilinear equations with $\mathcal{A}$-Laplace operators ${\rm div} \mathcal{A}(x, \nabla u)$ in place of $Δ_p$ are covered as well.
