Consider the Fucik eigenvalue problem
where $u = u^+  u^$, and $u^\pm := \max{\pm u, 0}$.
In [^{1}] it is proved that the first nontrivial curve of the Fucik spectrum can be described as a set of points $(s + c(s), c(s))$, where $s \in \mathbb{R}$ and $c(s)$ defined by
Here
where $S :=\{w \in W_0^{1,p}:~ \w\_{L^p}=1\}$ and $\varphi_1$ is the first eigenfunction.
There is another characterization of the first nontrivial curve of the Fucik spectrum. Namely, consider
Note that the admissible set for this minimization problem is nonempty for all $\beta > \lambda_1(p)$. This definition is, in essence, the same as of Theorem 1.2 in [^{2}] for the linear case $p=2$ (see also [^{3}]), and it was pointed out in that works that for $p>1$ this definition is also ok. Let us prove this fact explicitly.
Proposition. The set of points $(\alpha^*(\beta), \beta)$ is the first nontrivial curve of the Fucik spectrum.

Cuesta, M., De Figueiredo, D., Gossez, J. P. (1999). The beginning of the Fucik spectrum for the pLaplacian. Journal of Differential Equations, 159(1), 212238. ↩

Conti, M., Terracini, S., Verzini, G. (2005). On a class of optimal partition problems related to the Fucik spectrum and to the monotonicity formulae. Calculus of Variations and Partial Differential Equations, 22(1), 4572. ↩

Molle, R., Passaseo, D. (2015). Variational properties of the first curve of the Fucik spectrum for elliptic operators. Calculus of Variations and Partial Differential Equations, 54(4), 37353752. ↩