Lower bounds for eigenvalues of even ordered quasilinear differential equations

Abstract

In this paper, we establish lower bounds for the eigenvalues of even ordered quasilinear eigenvalue problems with indefinite weights. We first consider the case when the weight function is identically one and later generalize the results to any weight function within the p p -th Muckenhopt class. In the process, we obtain new Lyapunov-type inequalities for the corresponding problems and utilize them to obtain the lower bound of the k k -th eigenvalues. In particular, we show that the first eigenvalue is bounded below in terms of the integral of the weight instead of the integral of its positive part. Our results provide a much sharper bound than existing results.