On the Fučík spectrum of the p-Laplacian with no-flux boundary condition

In this paper, we study the quasilinear elliptic problem \begin{align*} \begin{aligned} -\Delta_{p} u&= a\l(u^+\r)^{p-1}-b\l(u^-\r)^{p-1} \quad && \text{in } \Omega,\\ u & = \text{constant} &&\text{on } \partial\Omega,\\ 0&=\int_{\partial \Omega}\left|\nabla u\right|^{p-2}\nabla u\cdot \nu \,\diff \sigma,&& \end{aligned} \end{align*} where the operator is the $p$-Laplacian and the boundary condition is of type no-flux. In particular, we consider the Fu\v{c}\'{\i}k spectrum of the $p$-Laplacian with no-flux boundary condition which is defined as the set $\fucik$ of all pairs $(a,b)\in\R^2$ such that the problem above has a nontrivial solution. It turns out that this spectrum has a first nontrivial curve $\mathcal{C}$ being Lipschitz continuous, decreasing and with a certain asymptotic behavior. Since $(\lambda_2,\lambda_2)$ lies on this curve $\mathcal{C}$, with $\lambda_2$ being the second eigenvalue of the corresponding no-flux eigenvalue problem for the $p$-Laplacian, we get a variational characterization of $\lambda_2$. This paper extends corresponding works for Dirichlet, Neumann, Steklov and Robin problems.