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October, 2020 Nonresonance and Resonance Problems for Nonlocal Elliptic Equations with Respect to the Fučik Spectrum
Sarika Goyal
Taiwanese J. Math. 24(5): 1153-1177 (October, 2020). DOI: 10.11650/tjm/200206

Abstract

In this article, we consider the following problem \[ \begin{cases} (-\Delta)^s u = \alpha u^+ - \beta u^{-} + f(u) + h &\textrm{in $\Omega$}, \\ u = 0 &\textrm{on $\mathbb{R}^n \setminus \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^n$ is a bounded domain with Lipschitz boundary, $n \gt 2s$, $0 \lt s \lt 1$, $(\alpha,\beta) \in \mathbb{R}^2$, $f \colon \mathbb{R} \to \mathbb{R}$ is a bounded and continuous function and $h \in L^2(\Omega)$. We prove the existence results in two cases: first, the nonresonance case where $(\alpha,\beta)$ is not an element of the Fučik spectrum. Second, the resonance case where $(\alpha,\beta)$ is an element of the Fučik spectrum. Our existence results follows as an application of the saddle point theorem. It extends some results, well known for Laplace operator, to the nonlocal operator.

Citation

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Sarika Goyal. "Nonresonance and Resonance Problems for Nonlocal Elliptic Equations with Respect to the Fučik Spectrum." Taiwanese J. Math. 24 (5) 1153 - 1177, October, 2020. https://doi.org/10.11650/tjm/200206

Information

Received: 30 May 2019; Revised: 29 January 2020; Accepted: 25 February 2020; Published: October, 2020
First available in Project Euclid: 4 March 2020

MathSciNet: MR4152661
Digital Object Identifier: 10.11650/tjm/200206

Subjects:
Primary: 35A15, 35B33, 35H39

Rights: Copyright © 2020 The Mathematical Society of the Republic of China

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Vol.24 • No. 5 • October, 2020
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