Electron. J. Differential Equations, Vol. 2019 (2019), No. 18, pp. 1-16.

Multiple solutions for discontinuous elliptic problems involving the fractional Laplacian

Jung-Hyun Bae, Yun-Ho Kim

Abstract:
In this article, we establish the existence of three weak solutions for elliptic equations associated to the fractional Laplacian
$$\displaylines{ (-\Delta)^s u = \lambda f(x,u) \quad \text{in } \Omega,\cr u= 0\quad \text{on } \mathbb{R}^N\setminus\Omega, }$$
where $\Omega$ is an open bounded subset in $\mathbb{R}^{N}$ with Lipschitz boundary, $\lambda$ is a real parameter, 0<s<1, N>2s, and $f:\Omega\times\mathbb{R} \to \mathbb{R}$ is measurable with respect to each variable separately. The main purpose of this paper is concretely to provide an estimate of the positive interval of the parameters $\lambda$ for which the problem above with discontinuous nonlinearities admits at least three nontrivial weak solutions by applying two recent three-critical-points theorems.

Submitted April 26, 2018. Published January 30, 2019.
Math Subject Classifications: 58E30, 49J52, 58E05.
Key Words: Fractional Laplacian; three-critical-points theorem; multiple solutions.

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Jung-Hyun Bae
Department of Mathematics
Sungkyunkwan University
Suwon 16419, Korea
email: hoi1000sa@skku.edu
Yun-Ho Kim
Department of Mathematics Education
Sangmyung University
Seoul 03016, Korea
email: kyh1213@smu.ac.kr

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