  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.

Analysis of PDEs