当前位置: X-MOL 学术Anal. Math. Phys. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Existence and non-existence results for fractional Kirchhoff Laplacian problems
Analysis and Mathematical Physics ( IF 1.7 ) Pub Date : 2021-06-12 , DOI: 10.1007/s13324-020-00435-7
Nemat Nyamoradi , Vincenzo Ambrosio

In this paper, we study the following fractional Kirchhoff-type problem:

$$\begin{aligned} \left[ a+b\Big (\iint _{{\mathbb {R}}^{2N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dx dy \Big )^{\theta -1}\right] (-\Delta )^s u= & {} |u|^{2^*_s- 2} u\\&\quad + \lambda f(x) |u|^{q-2}u, ~ in ~{\mathbb {R}}^N, \end{aligned}$$

where \((- \Delta )^s\) is the fractional Laplacian operator with \(0< s < 1\), \(\lambda \ge 0\), \(a \ge 0\), \(b> 0\), \(1<q<2\), \(N>2s\), and \(2^*_s= \frac{2 N}{N - 2s}\) is fractional critical Sobolev exponent. When \(\lambda =0\), under suitable values of the parameters \(\theta \), a and b, we obtain a non-existence result and the existence of infinitely many nontrivial solutions for the above problem. Also, for suitable weight function f(x), using the Nehari manifold technique and the fibbing maps, we prove the existence of at least two positive solutions for a sufficiently small choice of \(\lambda \).



中文翻译:

分数基尔霍夫拉普拉斯问题的存在和不存在结果

在本文中,我们研究以下分数基尔霍夫型问题:

$$\begin{aligned} \left[ a+b\Big (\iint _{{\mathbb {R}}^{2N}} \frac{|u(x)-u(y)|^2}{ |xy|^{N+2s}}dx dy \Big )^{\theta -1}\right] (-\Delta )^su= & {} |u|^{2^*_s- 2} u\ \&\quad + \lambda f(x) |u|^{q-2}u, ~ in ~{\mathbb {R}}^N, \end{aligned}$$

其中\((- \Delta )^s\)是分数拉普拉斯算子,其中\(0< s < 1\) , \(\lambda \ge 0\) , \(a \ge 0\) , \(b > 0\)\(1<q<2\)\(N>2s\)\(2^*_s= \frac{2 N}{N - 2s}\)是分数临界 Sobolev 指数。当\(\lambda =0\),在参数\(\theta \)ab 的合适值下,我们得到上述问题的不存在结果和无限多个非平凡解的存在。此外,对于合适的权重函数f ( x),使用 Nehari 流形技术和 fibbing 映射,我们证明了对于足够小的\(\lambda \)选择,至少存在两个正解。

更新日期:2021-06-13
down
wechat
bug