Electron. J. Differential Equations, Vol. 2016 (2016), No. 339, pp. 1-18.

Infinitely many solutions for Schrodinger-Kirchhoff type equations involving the fractional p-Laplacian and critical exponent

Li Wang, Binlin Zhang

Abstract:
In this article, we show the existence of infinitely many solutions for the fractional p-Laplacian equations of Schrodinger-Kirchhoff type equation
$$ M([u]_{s, p}^p) (-\Delta )_p^s u+V(x)|u|^{p-2}u= \alpha |u|^{ p_s^{*}-2 }u+\beta k(x)|u|^{q-2}u \quad x\in \mathbb{R}^N, $$
where $(-\Delta )^s_p$ is the fractional p-Laplacian operator, $[u]_{s,p}$ is the Gagliardo p-seminorm, $0 < s< 1<p<\infty$, $N> sp$, $1<q<p$, M is a continuous and positive function, V is a continuous and positive potential function and k(x) is a non-negative function in an appropriate Lebesgue space. By means of the concentration-compactness principle in fractional Sobolev space and Kajikiya's new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions which tend to zero for suitable positive parameters $\alpha$ and $\beta$.

Submitted October 11, 2016. Published December 30, 2016.
Math Subject Classifications: 35R11, 35A15, 47G20.
Key Words: Schrodinger-Kirchhoff type equation; fractional p-Laplacian; critical Sobolev exponent.

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Li Wang
School of Basic Science
East China Jiaotong University
Nanchang 330013, China
email: wangli.423@163.com
Binlin Zhang
Department of Mathematics
Heilongjiang Institute of Technology
Harbin 150050, China
email: zhangbinlin2012@163.com

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