Estimates for eigenvalues of quasilinear elliptic systems

Abstract

In this paper we introduce the generalized eigenvalues of a quasilinear elliptic system of resonant type. We prove the existence of infinitely many continuous eigencurves, which are obtained by variational methods. For the one-dimensional problem, we obtain an hyperbolic type function defining a region which contains all the generalized eigenvalues (variational or not), and the proof is based on a suitable generalization of Lyapunov's inequality for systems of ordinary differential equations. We also obtain a family of curves bounding by above the kth variational eigencurve.

Positive solutions of a weakly singular periodic eco-economic system with changing-sign perturbation

“…The first work dealing with generalized eigenvalues for p-Laplacian systems is due to Nápoli and Pinasco [8]. Inspired by that work, we present in this section some applications to fractional generalized eigenvalues related to problem (1.9)-(1.5).…”

“…In [8], Nápoli and Pinasco considered the quasilinear system of resonant type − (|u (t)| p−2 u (t)) = f(t)|u(t)| µ−2 |v(t)| ν u(t), − (|v (t)| q−2 v (t)) = g(t)|u(t)| µ |v(t)| Some nice applications to generalized eigenvalues are also presented in [8]. Inequality (1.6) was extended to more general problems by different authors (see [1,5,6,20,22] and references therein).…”

Lyapunov-type inequalities for Laplacian systems and applications to boundary value problems

“…Here, q + (t) := max{q(t), 0}, r > 0, γ > 1. In 2006, Napoli and Pinasco in [5] established Lyapunov-type inequalities for the quasilinear system of resonant type…”

Lyapunov-Type Inequalities for the Quasilinearq-Difference systems

“…In the paper, we consider boundary problem of the following quasilinear q-difference equation [10,11].…”