Infinitely Many Solutions for Fractional p-Kirchhoff Equations

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Abstract

In this paper we consider the existence of infinitely many weak solutions for fractional Schrödinger–Kirchhoff problems. Precisely speaking, we investigate
$$\begin{aligned} \left\{ \begin{array}{cl} M\left( \int _{\mathbb {R}^{2n}}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\mathrm{d}x\mathrm{d}y\right) (-\triangle )_p^su+V(x)|u|^{p-2}u=f(x,u), &{}\quad \mathrm{in}~\Omega ,\\ u=0, &{}\quad \mathrm{in}~\mathbb {R}^n\setminus \Omega , \end{array}\right. \end{aligned}$$
where \(\Omega \) is a bounded subset with Lipshcitz boundary \(\partial \Omega \), \(0<s<1\) is fixed, and \(1<p<n\), \((-\triangle )_p^s\) is the fractional p-Laplacian operator. Kirchhoff function M, potential function V and nonlinearity f satisfy some suitable assumptions. Under those conditions, some new multiplicity results are obtained by applying the fountain theorem and the dual fountain theorem.

Keywords

Schrödinger–Kirchhoff-type equation fractional p-Laplacian fountain theorem dual fountain theorem 

Mathematics Subject Classification

35R11 35A15 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of ScienceHohai UniversityNanjingPeople’s Republic of China
  2. 2.Faculty of Mathematics and PhysicsHuaiyin Institute of TechnologyHuai’anPeople’s Republic of China

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