论文标题
具有磁场和上部临界生长的退化分数Kirchhoff型系统
Degenerate fractional Kirchhoff-type system with magnetic fields and upper critical growth
论文作者
论文摘要
本文介绍了以下退化的分数Kirchhoff型系统，具有磁场和临界增长：$$ \ left \ {\ begin {array} {lll} - \ Mathfrak {m}（\ | U \ | _ {s，a}^2）[（ - δ）^s_au + u] = g_u（| x |，| u |^2，| u |^2，|^2） + \ lest（\ Mathcal {\ Mathcal {\ Mathcal {i} ＆\ \ mbox {in} \，\，\ \ mathbb {r}^n，\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \ left（\ Mathcal {i}_μ*| V |^{p^*} \ right）| V |^{p^* - 2} v \＆\ mbox {in} \，\，\，\，\ Mathbb {r} $$ white $ \ | u \ | _ {s，a} = \ left（\ iint _ {\ mathbb {r}^{2n}}} \ frac {| u（x）-e^{i^{i（x-y）\ cdot a（\ frac {x+y} {2}）} u（y）|^2} {| x-y |^{n+2s}} dx dy+\ in+\ in {\ mathbb {r}^n}^n}运算符和磁性电势。 $ \ Mathfrak {M}：\ Mathbb {r}^{+} _ {0} \ rightArrow \ Mathbb {r}^{+} _ 0 $是连续的Kirchhoff函数，$ \ natcal {n} $ c^1 $ - 功能$ g $满足某些合适的条件，$ p^* = \ frac {n+μ} {n-2s} $。我们使用极限索引理论证明了此问题的多重性结果。我们工作的新颖性是卷积术语和批判性非线性的出现。为了克服归因于基尔乔夫（Kirchhoff）功能和关键非线性造成的困难，我们引入了几种分析工具和分数版本浓度 - 紧凑性原理，这是证明紧凑条件的有用工具。
This paper deals with the following degenerate fractional Kirchhoff-type system with magnetic fields and critical growth: $$ \left\{ \begin{array}{lll} -\mathfrak{M}(\|u\|_{s,A}^2)[(-Δ)^s_Au+u] = G_u(|x|,|u|^2,|v|^2) + \left(\mathcal{I}_μ*|u|^{p^*}\right)|u|^{p^*-2}u \ &\mbox{in}\,\,\mathbb{R}^N,\\ \mathfrak{M}(\|v\|_{s,A})[(-Δ)^s_Av+v] = G_v(|x|,|u|^2,|v|^2) + \left(\mathcal{I}_μ*|v|^{p^*}\right)|v|^{p^*-2}v \ &\mbox{in}\,\,\mathbb{R}^N, \end{array}\right. $$ where $$\|u\|_{s,A}=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}dx dy+\int_{\mathbb{R}^N}|u|^2dx\right)^{1/2},$$ and $(-Δ)_{A}^s$ and $A$ are called magnetic operator and magnetic potential, respectively. $\mathfrak{M}:\mathbb{R}^{+}_{0}\rightarrow \mathbb{R}^{+}_0$ is a continuous Kirchhoff function, $\mathcal{I}_μ(x) = |x|^{N-μ}$ with $0<μ<N$, $C^1$-function $G$ satisfies some suitable conditions, and $p^* =\frac{N+μ}{N-2s}$. We prove the multiplicity results for this problem using the limit index theory. The novelty of our work is the appearance of convolution terms and critical nonlinearities. To overcome the difficulty caused by degenerate Kirchhoff function and critical nonlinearity, we introduce several analytical tools and the fractional version concentration-compactness principles which are useful tools for proving the compactness condition.
