On the characterization of the Dirichlet and Fucik spectra of the one-dimensional anisotropic p-Laplace operator

Abstract

The paper is concerned with the Dirichlet spectrum $\Lambda^{a,b}_p(0,L)$ of the anisotropic $p$-Laplace operator $- \Delta^{a,b}_{p}$ on an interval $(0,L)$ where \[ \Delta^{a,b}_p u:= \left(a^{p}[(u')^{+}]^{p-1}-b^{p}[(u')^{-}]^{p-1}\right)', \ \ a, b > 0. \] The set $\Lambda^{a,b}_p(0,L)$ and the respective eigenfunctions are completely characterized for $a \neq b$ in terms of the corresponding ones within the isotropic context. As an interesting application, we derive a new optimal Poincar\'e inequality that is stronger than the classical counterpart. The leading ideas are based on glue arguments of conveniently modified eigenfunctions and maximum type principles. More generally, our approach allows to characterize the Fu\v c\'ik spectrum $\Sigma^{a,b}_p(0,L)$ of $- \Delta^{a,b}_{p}$ on $(0,L)$ and mainly the corresponding solutions. All results are novelty even for the nonlinear operator $\Delta^{a,b}_2$.