Existence of the generalized Fu?ík spectrum for nonhomogeneous elliptic operators

摘要

By variational methods and Morse theory, we prove the existence of uncountably many (α, β) ∈ R~2 for which the equation ?div A(x,?u) = αu_+~(p?1) ?βu_-~(p?1) in ?, has a sign changing solution under the Neumann boundary condition, where a map A from ?×R~N to R~N satisfying certain regularity conditions. As a special case, the above equation contains the p-Laplace equation. However, the operator A is not supposed to be (p ? 1)-homogeneous in the second variable. In particular, it is shown that generally the Fu?ík spectrum of the operator ?div A(x,?u) on W~(1,p)(?) contains some open unbounded subset of R~2.