A Lyapunov type inequality for indefinite weights and eigenvalue homogenization (Q2790273)

The aim of the paper under review is the following quasilinear eigenvalue problem NEWLINE\[NEWLINE -(a(x)|u'|^{p-2}u')' = \lambda \rho(x)|u|^{p-2}u, NEWLINE\]NEWLINE for \(x \in (0,L)\) with the boundary conditions NEWLINE\[NEWLINE u(0) = u(L) = 0, NEWLINE\]NEWLINE where \( 1<p< \infty \), \( \lambda \) is a real parameter, \(\rho \in L^1([0,L]) \) and \(a(x)\) belongs to the Muckenhoupt class \(\mathcal{A}_p\). We have under suitable conditions two sequences of eigenvalues \(\{ \lambda_k^+ \}_{k\geq 1},\) \(\{ \lambda_k^- \}_{k\geq 1}\) going to \(\pm \infty\). For these eigenvalues the asymptotic behavior is known, but in order to obtain lower bounds for the eigenvalues another approach is necessary. The main results of the paper are such lower bounds. Let \(\lambda_k^\pm\) be the k-the eigenvalue of the problem then using variational arguments the following is provenNEWLINENEWLINENEWLINE\[NEWLINE \frac{k^{p-1}}{p} \bigg( \int_0^L a^{-\frac{1}{p-1}}(t) dt \bigg)^{1-p} \leq \lambda_k^\pm \sup_{(a,b) \subset [0,L]} \bigg| \int_a^b p(t) dt \bigg|. NEWLINE\]NEWLINENEWLINENEWLINEExtensions to higher order differential equations are also given.