On the Fu c ˘ ik spectrum for the pp-Laplacian with Robin boundary condition

The aim of this paper is to study the Fuc˘ik spectrum of the pp-Laplacian with Robin boundary condition given by −Δpu=a(u+)p−1−b(u−)p−1in Ω,|∇u|p−2∂u∂ν=−β|u|p−2uon ∂Ω, where β≥0β≥0. If β=0β=0, it reduces to the Fuc˘ik spectrum of the negative Neumannpp-Laplacian. The existence of a first nontrivial curve CC of this spectrum is shown and we prove some properties of this curve, e.g., CC is Lipschitz continuous, decreasing and has a certain asymptotic behavior. A variational characterization of the second eigenvalue λ2λ2 of the Robin eigenvalue problem involving the pp-Laplacian is also obtained.