We establish a generalization of Sturm–Picone comparison theorem for a pair of fractional nonlocal equations:

$$\begin{aligned} (-div. (A_1(x)\nabla ))^{s} u= & {} C_{1}(x) u \,\,\,\text { in }\,\,\Omega ,\\ u= & {} 0 \,\,\,\,\text { on }\,\,\,\,\,\partial \Omega , \end{aligned}$$

and

$$\begin{aligned} (-div. (A_2(x)\nabla ))^{s} v= & {} C_{2}(x) v \,\,\,\text { in}\,\,\Omega ,\\ v= & {} 0 \,\,\,\,\text { on}\,\,\,\,\,\partial \Omega , \end{aligned}$$

where \(\Omega \subset \mathbb {R}^n\) is an open bounded subset with smooth boundary, \(0<s<1,\,\,A_1,\,A_2\) are smooth, real symmetric and positive definite matrices on \({\overline{\Omega }}\) and \(C_{1}, C_{2}\in C^{\alpha }({\overline{\Omega }}).\)

中文翻译：

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分数非局部方程的Sturm-Picone定理

我们建立了几对分数阶非局部方程的Sturm-Picone比较定理的推广：

$$ \ begin {aligned}（-div。（A_1（x）\ nabla））^ {s} u =＆{} C_ {1}（x）u \，\，\，\ text {in} \， \，\ Omega，\\ u =＆{} 0 \，\，\，\，\ text {on} \，\，\，\，\，\ partial \ Omega，\ end {aligned} $$

和

$$ \ begin {aligned}（-div。（A_2（x）\ nabla））^ {s} v =＆{} C_ {2}（x）v \，\，\，\ text {in} \， \，\ Omega，\\ v =＆{} 0 \，\，\，\，\ text {on} \，\，\，\，\，\ partial \ Omega，\ end {aligned} $$

其中\（\ Omega \ subset \ mathbb {R} ^ n \）是具有光滑边界的开放边界子集，\（0 <s <1，\，\，A_1，\，A_2 \）是光滑，实对称且\（{\ overline {\ Omega}} \）和\（C_ {1}，C_ {2} \ in C ^ {\ alpha}（{\ overline {\ Omega}}）上的正定矩阵。