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

In this paper, we study the quasilinear elliptic problem −Δpu=au+p−1−bu−p−1inΩ,u=constanton∂Ω,0=∫∂Ω|∇u|p−2∇u⋅νdσ,where the operator is the p-Laplacian and the boundary condition is of type no-flux. In particular, we consider the Fučík spectrum of the p-Laplacian with no-flux boundary condition which is defined as the set Πp of all pairs (a,b)∈R2 such that the problem above has a nontrivial solution. It turns out that this spectrum has a first nontrivial curve C being Lipschitz continuous, decreasing and with a certain asymptotic behavior. Since (λ2,λ2) lies on this curve C, with λ2 being the second eigenvalue of the corresponding no-flux eigenvalue problem for the p-Laplacian, we get a variational characterization of λ2. This paper extends corresponding works for Dirichlet, Neumann, Steklov and Robin problems.