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 Fučík spectrum of the operator $-\mathrm{div}\, A(x, \nabla u)$ on $W^{1,p}(\Omega )$ contains some open unbounded subset of $\mathbb R ^2$.