# Journal of theKorean Mathematical SocietyJKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

## Article

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J. Korean Math. Soc. 2020; 57(6): 1451-1470

Published online November 1, 2020 https://doi.org/10.4134/JKMS.j190693

## Existence, multiplicity and regularity of solutions for the fractional $p$-Laplacian equation

Yun-Ho Kim

Sangmyung University

### Abstract

We are concerned with the following elliptic equations: \begin{equation*} \begin{cases} (-\Delta)_p^su=\lambda f(x,u) \quad \textmd{in} \ \ \Omega,\\ u= 0\quad \text{on}\ \ \mathbb{R}^N\backslash\Omega, \end{cases} \end{equation*} where $\lambda$ are real parameters, $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $0 Keywords: Fractional$p\$-Laplacian, weak solution, critical points, variational method

MSC numbers: 35R11, 35A15, 35J60, 49R05

Supported by: This research was supported by a 2017 Research Grant from Sangmyung University.

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