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2018 Existence of a weak solution for the fractional $p$-Laplacian equations with discontinuous nonlinearities via the Berkovits-Tienari degree theory
Yun-Ho Kim
Topol. Methods Nonlinear Anal. 51(2): 371-388 (2018). DOI: 10.12775/TMNA.2017.064

Abstract

We are concerned with the following nonlinear elliptic equations of the fractional $p$-Laplace type: \begin{equation*} \begin{cases} (-\Delta)_p^su \in \lambda[\underline{f}(x,u(x)), \overline{f}(x,u(x))] &\textmd{in } \Omega,\\ u= 0 &\text{on } \mathbb{R}^N\setminus\Omega, \end{cases} \end{equation*} where $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $\lambda$ is a parameter, $0 \lt s \lt 1\lt p\lt +\infty$, $sp \lt N$, and the measurable functions $\underline{f}$, $ \overline{f}$ are induced by a possibly discontinuous at the second variable function $f\colon \Omega\times\mathbb R \to \mathbb R$. By using the Berkovits-Tienari degree theory for upper semicontinuous set-valued operators of type (S$_+)$ in reflexive Banach spaces, we show that our problem with the discontinuous nonlinearity $f$ possesses at least one nontrivial weak solution. In addition, we show the existence of two nontrivial weak solutions in which one has negative energy and another has positive energy.

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Yun-Ho Kim. "Existence of a weak solution for the fractional $p$-Laplacian equations with discontinuous nonlinearities via the Berkovits-Tienari degree theory." Topol. Methods Nonlinear Anal. 51 (2) 371 - 388, 2018. https://doi.org/10.12775/TMNA.2017.064

Information

Published: 2018
First available in Project Euclid: 24 April 2018

zbMATH: 06928840
MathSciNet: MR3829036
Digital Object Identifier: 10.12775/TMNA.2017.064

Rights: Copyright © 2018 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.51 • No. 2 • 2018
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