Existence of the Fučík type spectrums for the generalized pp-Laplace operators

By variational methods and Morse theory, we prove the existence of uncountably many (α,β)∈R2(α,β)∈R2 for which the equation −divA(x,∇u)=αu+p−1−βu−p−1 in ΩΩ, has a sign changing solution under the Neumann boundary condition, where a map AA from Ω¯×RN to RNRN satisfies certain regularity conditions. As a special case, the above equation contains the pp-Laplace equation. However, the operator AA is not supposed to be (p−1)(p−1)-homogeneous in the second variable.