On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian

Abstract

We deal with the Dirichlet, Neumann and periodic eigenvalue problems for the equation $$ \quad(|u'|^{p-2}u')' +\lambda |u|^{q-2}u=0, \text{\quad on~~}(0,T), $$ where $T>0,$ $\lambda>0,$ and $p,q>1.$ For those problems we obtain a complete description of the spectra and a closed form representation of the corresponding eigenfunctions. As an application of our results we present sharp Poincar\'e and Wirtinger inequalities for the imbeddings $W_0^{1,p}(0,T)$ into $L^q(0,T)$ and $W_T^{1,p}(0,T)$ into $L^q(0,T),$ respectively.