On a variational characterization of a part of the Fučík spectrum and a superlinear equation for the Neumann $p$-Laplacian in dimension one

Abstract

In the first part of this paper a variational characterization of parts of the Fučík spectrum for the p-Laplacian in an interval is given. The proof uses a linking theorem on suitably constructed sets in $W^{1,p}(0,1)$. In the second part, a superlinear equation with Neumann boundary conditions on an interval is considered, where the nonlinearity intersects all but the first eigenvalues. It is proved that under certain conditions this equation is solvable for arbitrary forcing terms. The proof uses a comparison of the minimax levels of the functional associated to this equation with suitable minimax values related to the Fučík spectrum.