On the Fučík spectrum of the [formula omitted]-Laplacian with no-flux boundary condition.

In this paper, we study the quasilinear elliptic problem − Δ p u = a u + p − 1 − b u − p − 1 in Ω , u = constant on ∂ Ω , 0 = ∫ ∂ Ω | ∇ u | p − 2 ∇ u ⋅ ν d σ , 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 Π p of all pairs (a , b) ∈ R 2 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 (λ 2 , λ 2) lies on this curve C , with λ 2 being the second eigenvalue of the corresponding no-flux eigenvalue problem for the p -Laplacian, we get a variational characterization of λ 2. This paper extends corresponding works for Dirichlet, Neumann, Steklov and Robin problems. [ABSTRACT FROM AUTHOR]