Electron. J. Differential Equations, Vol. 2018 (2018), No. 74, pp. 1-21.

First curve of Fucik spectrum for the p-fractional Laplacian operator with nonlocal normal boundary conditions

Divya Goel, Sarika Goyal, Konijeti Sreenadh

Abstract:
In this article, we study the Fucik spectrum of the p-fractional Laplace operator with nonlocal normal derivative conditions which is defined as the set of all $(a,b)\in\mathbb{R}^2$ such that
$$\displaylines{ \Lambda_{n,p}(1-\alpha)(-\Delta)_{p}^{\alpha} u + |u|^{p-2}u = \frac{\chi_{\Omega_\epsilon}}{\epsilon} (a (u^{+})^{p-1} - b (u^{-})^{p-1}) \quad \text{in }\Omega, \cr \mathcal{N}_{\alpha,p} u = 0 \quad \text{in }\mathbb{R}^n \setminus \overline{\Omega}, }$$
has a non-trivial solution u, where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with Lipschitz boundary, $p \geq 2$, $n>p \alpha$, $\epsilon, \alpha \in(0,1)$ and $\Omega_{\epsilon}:=\{x \in \Omega: d(x,\partial \Omega) \leq \epsilon \}$. We show existence of the first non-trivial curve $\mathcal{C}$ of the Fucik spectrum which is used to obtain the variational characterization of a second eigenvalue of the problem defined above. We also discuss some properties of this curve $\mathcal{C}$, e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior and non-resonance with respect to the Fucik spectrum.

Submitted November 22, 2017. Published March 17, 2018.
Math Subject Classifications: 35A15, 35J92, 35J60.
Key Words: Nonlocal operator; Fucik spectrum; Steklov problem; Non-resonance.

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Divya Goel
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khas, New Delhi-110016, India
email: divyagoel2511@gmail.com
Sarika Goyal
Department of Mathematics
Bennett University, Greater Noida
Uttar Pradesh - 201310, India
email: sarika1.iitd@gmail.com
Konijeti Sreenadh
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khaz, New Delhi-110016, India
email: sreenadh@maths.iitd.ac.in

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