Existence, Nonexistence, and Multiple Results for the Fractional p-Kirchhoff-type Equation in equation id-i-eq1"> equation-source format-t-e-x" xmlns:search="http://marklogic.com/appservices/search">\({\mathbb{R}^N}\)

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• 作者：Caisheng Chen ; Yunfeng Wei
• 关键词：Fractional p ; Laplacian ; Schrödinger–Kirchhoff ; type equation ; Variational methods
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：13
• 期：6
• 页码：5077-5091
• 全文大小：603 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1660-5454
• 卷排序：13

文摘

In this paper, we investigate the fractional p-Kirchhoff-type equation$$\begin{array}{ll} M\left(\int\int\limits_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\right)(-\Delta)_p^su+V(x)|u|^{p-2}u+b(x)|u|^{q-2}u\\ \quad = \lambda a(x)|u|^{m-2}u, \; x\in\mathbb{R}^N,\end{array}$$where \({\lambda}\) is a real parameter, \({(-\Delta)_p^s }\) is the fractional p-Laplacian operator with \({0 < s < 1 < p}\) and \({ps < N}\), \({V(x),a(x), b(x): \mathbb{R}^N\to (0,\infty)}\) are three positive weights, and M is a continuous and positive function. The case \({1 < q < m < p_s^*}\) is considered. Using variational methods, we prove the existence, nonexistence, and multiplicity of solutions for the above equation depending on \({\lambda, m,q}\) and according to the weight functions a(x) and b(x). Our results extend the previous works of Pucci et al. [23] and of Xiang et al. [29].KeywordsFractional p-LaplacianSchrödinger–Kirchhoff-type equationVariational methodsThis work was supported by the Fundamental Research Funds for the Central Universities of China (2015B31014).