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
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
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