Electron. J. Differential Equations, Vol. 2017 (2017), No. 208, pp. 1-14.

Existence of infinitely many solutions for fractional p-Laplacian equations with sign-changing potential

Youpei Zhang, Xianhua Tang, Jian Zhang

Abstract:
In this article, we prove the existence of infinitely many solutions for the fractional $p$-Laplacian equation
$$ (-\Delta)^s_p u+V(x)|u|^{p-2}u=f(x,u),\quad x\in \mathbb{R}^N $$
where $s\in(0,1)$, $2\leq p<\infty$. Based on a direct sum decomposition of a space $E^s$, we investigate the multiplicity of solutions for the fractional p-Laplacian equation in $\mathbb{R}^N$. The potential V is allowed to be sign-changing, and the primitive of the nonlinearity f is of super-p growth near infinity in u and allowed to be sign-changing. Our assumptions are suitable and different from those studied previously.

Submitted January 25, 2017. Published September 8, 2017.
Math Subject Classifications: 35J60, 35J20.
Key Words: Fractional p-Laplacian; multiple solutions; variational methods; sign-changing potential.

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Youpei Zhang
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
email: zhangypzn@163.com
Xianhua Tang
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
email: tangxh@mail.csu.edu.cn
Jian Zhang
School of Mathematics and Statistics
Hunan University of Commerce
Changsha, 410205 Hunan, China
email: zhangjian433130@163.com

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