Applied Mathematics and Optimization ( IF 1.8 ) Pub Date : 2021-03-26 , DOI: 10.1007/s00245-021-09763-x Wen Guan , Hai-Feng Huo
In this paper, we study the following fractional Kirchhoff-type equation
$$\begin{aligned}{\left\{ \begin{array}{ll} -(a+ b\int _{{\mathbb {R}}^{N}}\int _{{\mathbb {R}}^{N}}|u(x)-u(y)|^{2}K(x-y)dxdy){\mathcal {L}}_{K}u=|u|^{2_{\alpha }^{*}-2}u+\mu f(u), ~\ x\in \Omega ,\\ u=0, ~\ x\in {\mathbb {R}}^{N}\backslash \Omega , \end{array}\right. } \end{aligned}$$where \(\Omega \subset {\mathbb {R}}^{N}\) is a bounded domain with a smooth boundary, \(\alpha \in (0,1)\), \(2\alpha<N<4\alpha \), \(2_{\alpha }^{*}\) is the fractional critical Sobolev exponent and \(\mu , a, b>0\); \({\mathcal {L}}_{K}\) is non-local integrodifferential operator. Under suitable conditions on f, for \(\mu \) large enough, by using constraint variational method and the quantitative deformation lemma, we obtain a ground state sign-changing (or nodal) solution to this problem, and its energy is strictly larger than twice that of the ground state solutions.
中文翻译：
临界增长分数阶Kirchhoff型方程基态改变解的存在性
在本文中，我们研究以下分数基尔霍夫型方程
$$ \ begin {aligned} {\ left \ {\ begin {array} {ll}-（a + b \ int _ {{\ mathbb {R}} ^ {N}} \ int _ {{\ mathbb {R} } ^ {N}} | u（x）-u（y）| ^ {2} K（xy）dxdy）{\数学{L}} _ {K} u = | u | ^ {2 _ {\ alpha} ^ {**-2} u + \ mu f（u），〜\ x \ in \ Omega，\\ u = 0，〜\ x \ in {\ mathbb {R}} ^ {N} \反斜杠\ Omega， \ end {array} \ right。} \ end {aligned} $$其中\（\ Omega \ subset {\ mathbb {R}} ^ {N} \）是具有光滑边界的有界域\（\ alpha \ in（0,1）\），\（2 \ alpha <N <4 \ alpha \），\（2 _ {\ alpha} ^ {*} \）是分数临界Sobolev指数和\（\ mu，a，b> 0 \） ; \（{\ mathcal {L}} _ {K} \）是非本地积分微分运算符。在f的适当条件下，对于\（\ mu \）足够大，通过使用约束变分方法和定量变形引理，我们获得了该问题的基态符号改变（或节点）解，并且其能量严格较大是基态解决方案的两倍。
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