Abstract

In this work we study the sequence of variational eigenvalues of a system of resonant type involving p- and q-Laplacians on Ω ⊂ R N , with a coupling term depending on two parameters α and β satisfying α / p + β / q = 1 . We show that the order of growth of the kth eigenvalue depends on α + β , λ k = O ( k α + β N ) .

Highlights

  • Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω, the s−laplacian operator is ∆su = div(|∇u|s−2∇u), the exponents satisfy 1 < p, q < +∞, and the positive parameters α, β satisfy

  • Up to our knowledge, the first work where this problem was addressed was [12] where we obtained a generalization of the Lyapunov inequality together with an upper bound of the variational eigenvalues in terms of the ones of a single p-laplacian equation for the one dimensional case

  • We show that λk(p, q) ≤ cλk(α + β), for a fixed constant c depending only on α, β, and Ω

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Summary

Introduction

This paper is devoted to the study of the asymptotic behavior of eigenvalues of resonant quasilinear systems (1.1). Up to our knowledge, the first work where this problem was addressed was [12] where we obtained a generalization of the Lyapunov inequality together with an upper bound of the variational eigenvalues in terms of the ones of a single p-laplacian equation for the one dimensional case. In [12] we obtained an upper bound of the first eigenvalue of the one dimensional problem (1.4) in terms of the first eigenvalue of the single p-laplacian equation.

Some known facts
Estimates for the Spectral Counting Function
A Lower Bound for the First Eigenvalue
An Upper Bound for the First Eigenvalue
Findings
An explicit lower bound on the Main Theorem
Full Text
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