论文标题
非线性系统的阳性解决方案的先验估计和存在
The A Priori Estimate and Existence of the Positive Solution for A Nonlinear System Involving the Fractional Laplacian
论文作者
论文摘要
在论文中，我们考虑了分数椭圆系统\ begin {equation*} \ left \ {\ oken {arnay} {ll} {ll}（ - δ）^{\ frac {α_1} {2} {2} {2}} {2}} {x） u} {\ partial x_i}+b（x）u（x）= f（x，x，u，v），＆\ mbox {in}ω，\\（ - δ） v} {\ partial x_i}+c（x）v（x）= g（x，x，u，v），＆\ mbox {in}ω，\\ u = v = 0，＆\ mbox {in} \ mathbb {in} \ end {equation*}其中$ω$是一个有界域，$ c^2 $ boundare in $ \ mathbb {r}^n $和$ n> \ max \ max \ {α_1，α_2\} $。我们首先利用爆炸和重新缩放方法在$ 1 <α_1，α_2<2 $时得出正溶液的先验估计。然后以$ 0 <α_1，α_2<1 $，我们获得了正溶液的规律性估计。最重要的是，使用拓扑度理论，我们证明了阳性解决方案的存在。
In the paper, we consider the fractional elliptic system \begin{equation*}\left\{\begin{array}{ll} (- Δ)^{\frac{α_1}{2}}u(x)+\sum\limits^n_{i=1}b_i(x)\frac{\partial u}{\partial x_i}+B(x)u(x)=f(x,u,v),& \mbox { in } Ω,\\ (- Δ)^{\frac{α_2}{2}}v(x)+\sum\limits^n_{i=1}c_i(x)\frac{\partial v}{\partial x_i}+C(x)v(x)=g(x,u,v),& \mbox { in } Ω,\\ u=v=0, & \mbox { in } \mathbb{R}^n\setminusΩ, \end{array} \right.\label{a-1.2} \end{equation*} where $Ω$ is a bounded domain with $C^2$ boundary in $\mathbb{R}^n$ and $n>\max\{α_1,α_2\}$. We first utilize the blowing-up and re-scaling method to derive the a priori estimate for positive solutions when $1<α_1,α_2 <2$. Then for $0<α_1,α_2 <1$, we obtain the regularity estimate of positive solutions. On top of this, using the topological degree theory we prove the existence of positive solutions.
