Precise asymptotic of eigenvalues of resonant quasilinear systems

TL;DR

This paper derives the precise asymptotic behavior of eigenvalues for a resonant quasilinear system involving p- and q-Laplacians, revealing how eigenvalue growth depends on system parameters and dimension.

Contribution

It establishes the asymptotic order of eigenvalues for a coupled p- and q-Laplacian system with parameter-dependent resonance, extending spectral theory in nonlinear PDEs.

Findings

Eigenvalues grow as O(k^{(ppa+eta)/N})

Growth rate depends on parameters ppa, eta, and dimension N

Provides explicit asymptotic estimates for eigenvalues in resonant systems

Abstract

In this work we study the sequence of variational eigenvalues of a system of resonant type involving $p-$ and $q-$laplacians on $\Omega \subset \R^N$, with a coupling term depending on two parameters $\alpha$ and $\beta$ satisfying $\alpha/p + \beta/q = 1$. We show that the order of growth of the $k^{th}$ eigenvalue depends on $\alpha+\beta$, $\lam_k = O(k^{\frac{\alpha+\beta}{N}})$.