On the closed solution to some nonhomogeneous eigenvalue problem with \(p\)-Laplacian. (English) Zbl 1015.34071

The authors investigate various boundary value problems associated with the equation \[ (|u'|^{p-2}u')'+\lambda |u|^{q-2}u=0,\quad t\in (0,T), \quad p,q>1. \tag{\(*\)} \] A complete description of the spectrum as well as a closed form representation of the corresponding eigenfunctions are obtained. As an application, the results presented in the paper give the sharp Poincaré and Wirtinger inequalities for the imbeddings \(W^{1,p}_0(0,T)\) into \(L^q(0,T)\) and \(W^{1,p}_T(0,T)\) into \(L^q(0,t)\). The results of the paper are based on a detailed analysis of the solutions to \((*)\) via certain elliptic-type integrals.


34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B15 Nonlinear boundary value problems for ordinary differential equations