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Sign-Changing Solutions for Fractional Kirchhoff-Type Equations with Critical and Supercritical Nonlinearities
Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2021-03-18 , DOI: 10.1007/s00009-021-01733-5
Liu Gao , Chunfang Chen , Jianhua Chen , Chuanxi Zhu

This paper concerns the existence of sign-changing solutions for the following fractional Kirchhoff-type equation with critical and supercritical nonlinearities

$$\begin{aligned} \left( a+b[u]^{2}\right) (-\Delta )^{\alpha }u+V(x)u=f(x,u)+\lambda |u| ^{r-2}u,\,\, \text {in}\,\,\mathbb {R}^3, \end{aligned}$$

where \(a,b>0\) are constants, \(\alpha \in (\frac{3}{4}, 1)\), \(\lambda >0\) is a real parameter, \(r\ge 2_{\alpha }^{*}=\frac{6}{3-2\alpha }\), \((-\Delta )^{\alpha }\) is the fractional Laplace operator, the potential function V and the nonlinearity f satisfy some suitable conditions. By combining an appropriate truncation argument with Moser iteration method, we prove that the existence of sign-changing solutions for the above equation when the parameter \(\lambda \) is sufficiently small. Our results enrich and improve the previous ones in the literature.



中文翻译:

具有临界和超临界非线性分数阶Kirchhoff型方程的符号转换解

本文关注以下具有临界和超临界非线性分数阶Kirchhoff型方程的正变解的存在性

$$ \ begin {aligned} \ left(a + b [u] ^ {2} \ right)(-\ Delta)^ {\ alpha} u + V(x)u = f(x,u)+ \ lambda | u | ^ {r-2} u,\,\,\ text {in} \,\,\ mathbb {R} ^ 3,\ end {aligned} $$

其中\(a,b> 0 \)是常量,\(\ alpha \ in(\ frac {3} {4},1)\)\(\ lambda> 0 \)是实数,\(r \ ge 2 _ {\ alpha} ^ {*} = \ frac {6} {3-2 \ alpha} \)\((-Delta ^^ {\ alpha} \)是分数拉普拉斯算子,势函数V和非线性f满足一些合适的条件。通过将适当的截断参数与Moser迭代方法相结合,我们证明了当参数\(\ lambda \)足够小时,上述方程的符号转换解的存在。我们的结果丰富并改进了文献中的先前结果。

更新日期:2021-03-19
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