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On the de Thélin eigenvalue problem and Landesman-Lazer conditions for quasilinear systems

David Arcoya, Natalino Borgia, Silvia Cingolani

TL;DR

This work studies a quasilinear de Thélin eigenproblem for a coupled p– and q–Laplacian system under resonance, introducing a robust variational framework with Φ, Ψ, and the manifold 𝔐 to define a spectrum {λ_k}. It proves the first eigenvalue λ1 is isolated and simple, and it develops a deformation-based approach on C^1 submanifolds to generate a full sequence of eigenvalues, recovering λ1 via a minimax characterization. Furthermore, it demonstrates the existence of a weak solution at resonance around λ1 under Landesman–Lazer type conditions and obtains a second solution via a careful energy geometry analysis of J, extending previous resonance results for quasilinear systems. The results collectively advance the spectral theory for coupled p– and q–Laplacian systems and provide new tools for nonlinear resonance phenomena in this setting.

Abstract

In this paper we prove that the smallest eigenvalue $λ_1$ of the eigenvalue problem for a quasilinear elliptic systems introduced by de Thélin in \cite{DT}, is not only simple (in a suitable sense), but also isolated. Moreover, we characterize variationally a sequence $\{λ_k\}_k$ of eigenvalues, taking into account a suitable deformation lemma for $C^1$ submanifolds proved in \cite{BON}. Furthermore we prove the existence of a weak solution for a quasilinear elliptic systems in resonance around $λ_1$, under new sufficient Landesman-Lazer type conditions, extending the results by Arcoya and Orsina \cite{AO}.

On the de Thélin eigenvalue problem and Landesman-Lazer conditions for quasilinear systems

TL;DR

This work studies a quasilinear de Thélin eigenproblem for a coupled p– and q–Laplacian system under resonance, introducing a robust variational framework with Φ, Ψ, and the manifold 𝔐 to define a spectrum {λ_k}. It proves the first eigenvalue λ1 is isolated and simple, and it develops a deformation-based approach on C^1 submanifolds to generate a full sequence of eigenvalues, recovering λ1 via a minimax characterization. Furthermore, it demonstrates the existence of a weak solution at resonance around λ1 under Landesman–Lazer type conditions and obtains a second solution via a careful energy geometry analysis of J, extending previous resonance results for quasilinear systems. The results collectively advance the spectral theory for coupled p– and q–Laplacian systems and provide new tools for nonlinear resonance phenomena in this setting.

Abstract

In this paper we prove that the smallest eigenvalue of the eigenvalue problem for a quasilinear elliptic systems introduced by de Thélin in \cite{DT}, is not only simple (in a suitable sense), but also isolated. Moreover, we characterize variationally a sequence of eigenvalues, taking into account a suitable deformation lemma for submanifolds proved in \cite{BON}. Furthermore we prove the existence of a weak solution for a quasilinear elliptic systems in resonance around , under new sufficient Landesman-Lazer type conditions, extending the results by Arcoya and Orsina \cite{AO}.
Paper Structure (6 sections, 10 theorems, 167 equations)

This paper contains 6 sections, 10 theorems, 167 equations.

Table of Contents

  1. INTRODUCTION
  2. SIMPLICITY OF $\lambda_1$
  3. ISOLATION OF $\lambda_1$
  4. A SEQUENCE OF EIGENVALUES
  5. PALAIS-SMALE CONDITION OF $J$
  6. GEOMETRY OF $J$

Key Result

Theorem 1.1

The set of all eigenfunctions associated to $\lambda_1$ is given by where $(\varphi_0,\psi_0)$ is the unique nontrivial eigenfunction satisfying $(\varphi_0,\psi_0) \in \Sigma$, $\varphi_0 >0$ and $\psi_0>0$ in $\Omega$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • Proposition 4.3
  • ...and 6 more