On the first curve of Fu\v{c}ik Spectrum Of $p$-fractional Laplacian Operator with nonlocal normal boundary conditions

TL;DR

This paper investigates the Fučík spectrum of the p-fractional Laplacian with nonlocal boundary conditions, establishing the existence and properties of the first non-trivial spectral curve and its implications for eigenvalues.

Contribution

It introduces the first non-trivial spectral curve for the Fučík spectrum of the p-fractional Laplacian with nonlocal boundary conditions, providing variational characterization and analyzing its properties.

Findings

Existence of the first non-trivial spectral curve or the Fude9k spectrum.

The spectral curve is Lipschitz continuous, strictly decreasing, and has specific asymptotic behavior.

The spectral curve exhibits nonresonance with respect to the Fude9k spectrum.

Abstract

In this article, we study the Fu\v{c}ik spectrum of the $p$-fractional Laplace operator with nonlocal normal derivative conditions which is defined as the set of all $(a,b)\in \mb R^2$ such that $$ \mc (F_p)\left\{ \begin{array}{lr} \Lambda_{n,p}(1-\al)(-\Delta)_{p}^{\al} u + |u|^{p-2}u = \frac{\chi_{\Omega_\e}}{\e} (a (u^{+})^{p-1} - b (u^{-})^{p-1}) \;\quad \text{in}\; \Omega,\quad \\ \mc{N}_{\al,p} u = 0 \; \quad \mbox{in}\; \mb R^n \setminus \overline{\Omega}, \end{array} \right. $$ has a non-trivial solution $u$, where $\Omega$ is a bounded domain in $\mb R^n$ with Lipschitz boundary, $p \geq 2$, $n>p \al $, $\e, \al \in(0,1)$ and $\Omega{_\e}:=\{x \in \Omega: d(x,\pa \Omega)\leq \e \}$. We showed existence of the first non-trivial curve $\mc C$ of this spectrum which is used to obtain the variational characterization of a second eigenvalue of the problem $\mc (F_p)$. We also discuss some properties of this curve $\mc C$, e.g. Lipschitz continuous, strictly decreasing and asymptotic behaviour and nonresonance with respect to the Fu\v{c}ik spectrum.