Taiwanese Journal of Mathematics

Nonresonance and Resonance Problems for Nonlocal Elliptic Equations with Respect to the Fučik Spectrum

Sarika Goyal

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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.

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Taiwanese J. Math., Advance publication (2020), 25 pages.

First available in Project Euclid: 4 March 2020

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Primary: 35A15: Variational methods 35B33: Critical exponents 35H39

nonlocal problem Fučik spectrum resonance nonresonance saddle point theorem


Goyal, Sarika. Nonresonance and Resonance Problems for Nonlocal Elliptic Equations with Respect to the Fučik Spectrum. Taiwanese J. Math., advance publication, 4 March 2020. doi:10.11650/tjm/200206. https://projecteuclid.org/euclid.twjm/1583290813

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