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Existence and asymptotic behavior of sign-changing solutions for fractional Kirchhoff-type problems in low dimensions

  • Sitong Chen
  • Xianhua Tang
  • Fangfang Liao
Article

Abstract

This paper is dedicated to studying the following fractional Kirchhoff-type equation
$$\begin{aligned} \left( a+b\int _{\mathbb {R}^N}|(-\triangle )^{\alpha /2}{u}|^2\mathrm {d}x\right) (-\triangle )^{\alpha }u+V(|x|)u =f(|x|, u), \ \ \ \ x\in {\mathbb {R}}^{N}, \end{aligned}$$
where \(a, b>0\), either \(N=2\) and \(\alpha \in (1/2,1)\) or \(N=3\) and \(\alpha \in (3/4,1)\) holds, \(V\in \mathcal {C}(\mathbb {R}^{N}, [0,\infty ))\) and \(f\in \mathcal {C}(\mathbb {R}^N\times \mathbb {R}, \mathbb {R})\). By combining the constraint variational method with some new inequalities, we prove that the above problem possesses a radial sign-changing solution \(u_b\) for \(b\ge 0\) without the usual Nehari-type monotonicity condition on f, and its energy is strictly larger than twice that of the ground state radial solutions of Nehari-type. Moreover, we establish the convergence property of \(u_b\) as \(b\searrow 0\). In particular, our results unify both asymptotically cubic and super-cubic cases, which improve and complement the existing ones in the literature.

Keywords

Fractional Kirchhoff type problems Sign-changing solution Asymptotic behavior 

Mathematics Subject Classification

35J20 35Q55 35Q51 53C35 

Notes

Acknowledgements

This work was supported by the NNSF (11701487, 11626202), Hunan Provincial Natural Science Foundation of China (2016JJ6137), Scientific Research Fund of Hunan Provincial Education Department (15B223). The authors thank the anonymous referees for their valuable suggestions and comments.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaPeople’s Republic of China
  2. 2.Department of MathematicsXiangnan UniversityChenzhouPeople’s Republic of China

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