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

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.

Keywords

• Nonlocal operator
• Fucik spectrum
• Steklov problem
• Non-resonance