Existence of the generalized Fučík spectrum for nonhomogeneous elliptic operators

Abstract

By variational methods and Morse theory, we prove the existence of uncountably many $$(\alpha ,\beta )\in \mathbb R ^2$$ for which the equation $$-\mathrm{div}\, A(x, \nabla u)=\alpha u_+^{p-1} -\beta u_-^{p-1}$$ in $$\Omega $$ , has a sign changing solution under the Neumann boundary condition, where a map $$A$$ from $$\overline{\Omega }\times \mathbb R ^N$$ to $$\mathbb R ^N$$ 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 Fucik spectrum of the operator $$-\mathrm{div}\, A(x, \nabla u)$$ on $$W^{1,p}(\Omega )$$ contains some open unbounded subset of $$\mathbb R ^2$$ .