论文标题
具有关键非线性的分数椭圆系统
Fractional elliptic systems with critical nonlinearities
论文作者
论文摘要
在本文中，我们研究了以下非本地方程系统的积极解决方案：\ begin {qoration*} \ left \ {\ begin {aligned} ＆（δ） ＆（ - δ）^s v = \fracβ{2_s^*} | v |^{β-2} v | u | u |^α+g（x）\; \; \; \ text {in} ＆\ qquad u，\，v> 0 \，\ mbox {in} \，\ mathbb {r}^{n}， \ end {Aligned} \正确的。 \ end {等式*}其中$ n> 2s $，$α，\，\，β> 1 $，$α+β= 2n/（n-2s）$，$ f，\，g $是$ \ dot {h} s（h} s（h} s（h} s（h} s（\ mathbbbbbbbbb，\ mathbbbb {r r} n}），当$ f = 0 = g $时，我们表明上述系统的基态解决方案为{\ it唯一}。另一方面，当$ f $和$ g $是具有ker $（f）$ = ker $（g）$的非平凡的非负函数时，我们确定上述系统的至少两种不同的积极解决方案，前提$ \ | g \ | _ {（\ dot {h}^s）'} $足够小。此外，我们还提供了一个全球的紧凑性结果，从而完整描述了上述系统的palais-smale序列。
In this paper we study positive solutions to the following nonlocal system of equations: \begin{equation*} \left\{\begin{aligned} &(-Δ)^s u = \fracα{2_s^*}|u|^{α-2}u|v|^β+f(x)\;\;\text{in}\;\mathbb{R}^{N}, &(-Δ)^s v = \fracβ{2_s^*}|v|^{β-2}v|u|^α+g(x)\;\;\text{in}\;\mathbb{R}^{N}, & \qquad u, \, v >0\, \mbox{ in }\,\mathbb{R}^{N}, \end{aligned} \right. \end{equation*} where $N>2s$, $α,\,β>1$, $α+β=2N/(N-2s)$, and $f,\, g$ are nonnegative functionals in the dual space of $\dot{H}^s(\mathbb{R}^{N})$. When $f=0=g$, we show that the ground state solution of the above system is {\it unique}. On the other hand, when $f$ and $g$ are nontrivial nonnegative functionals with ker$(f)$=ker$(g)$, then we establish the existence of at least two different positive solutions of the above system provided that $\|f\|_{(\dot{H}^s)'}$ and $\|g\|_{(\dot{H}^s)'}$ are small enough. Moreover, we also provide a global compactness result, which gives a complete description of the Palais-Smale sequences of the above system.
