Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Eigenvalue estimates and maximum principle for Lane-Emden systems, and applications to poly-Laplacian equations

Sabri Bahrouni, Edir Júnior Ferreira Leite, Gustavo Ferron Madeira

Abstract

This paper deals with explicit upper and lower bounds for principal eigenvalues and the maximum principle associated to generalized Lane-Emden systems (GLE systems, for short). Regarding the bounds, we generalize the upper estimate of Berestycki, Nirenberg and Varadhan [Comm. Pure Appl. Math. (1994), 47-92] for the first eigenvalue of linear scalar problems on general domains to the case of strongly coupled GLE systems with $m \geqslant 2$ equations on smooth domains. The explicit lower estimate we obtain is also used to derive a maximum principle to GLE systems relying in terms of quantitative ingredients. Furthermore, as applications of the previous results, upper and lower estimates for the first eigenvalue of weighted poly-Laplacian eigenvalue problems with $L^p$ weights $(p>n)$ and Navier boundary condition are obtained. Moreover, a strong maximum principle depending on the domain and the weight function for scalar problems involving the poly-Laplacian operator is also established.

Eigenvalue estimates and maximum principle for Lane-Emden systems, and applications to poly-Laplacian equations

Abstract

This paper deals with explicit upper and lower bounds for principal eigenvalues and the maximum principle associated to generalized Lane-Emden systems (GLE systems, for short). Regarding the bounds, we generalize the upper estimate of Berestycki, Nirenberg and Varadhan [Comm. Pure Appl. Math. (1994), 47-92] for the first eigenvalue of linear scalar problems on general domains to the case of strongly coupled GLE systems with equations on smooth domains. The explicit lower estimate we obtain is also used to derive a maximum principle to GLE systems relying in terms of quantitative ingredients. Furthermore, as applications of the previous results, upper and lower estimates for the first eigenvalue of weighted poly-Laplacian eigenvalue problems with weights and Navier boundary condition are obtained. Moreover, a strong maximum principle depending on the domain and the weight function for scalar problems involving the poly-Laplacian operator is also established.

Paper Structure

This paper contains 8 sections, 8 theorems, 94 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications to poly-Laplacian equations
  4. Proof of Theorem \ref{['teo1']} and Corollary \ref{['EQ1']}
  5. Proof of Theorem \ref{['LE']}
  6. Proof of Theorem \ref{['upper']}
  7. Proof of Theorem \ref{['teo2']}
  8. Bounds for eigenvalues of (\ref{['poly']}) and proof of Theorems \ref{['teo4']} and \ref{['cor3']}

Key Result

Theorem 1.1

($L^\infty$-weights) Let $\rho \in L^\infty(\Omega;\Bbb{R}^m)$. Suppose that $\Omega$ satisfies and hip6 for $i = 1,\ldots,m$ and let ${\bf \Lambda_1}$ be the principal $(m-1)$-hypersurface (hyp). Assume also that $\alpha_i=1$ for all $i = 1,\ldots,m$ (i.e., linear case). Then for $\Lambda_0=(\lambda_{01},\ldots,\lambda_{0m})\in {\bf \Lambda_1}$, we obtain Moreover, if in addition $\Omega$ satis

Figures (2)

  • Figure 1: Aspect of the principal $2$-hypersurface ${\bf \Lambda_1}=\left\{(\lambda_1,\lambda_2,\lambda_3) \in (0, \infty)^3 :\, \lambda_1 \lambda^{\alpha_1}_2\lambda^{\alpha_1\alpha_{2}}_3 = \lambda_*\right\}$.
  • Figure 2: For $m=2$, ${\bf \Lambda_1}=\left\{(\lambda_1,\lambda_2) \in (0, \infty)^2 :\, \lambda_1 \lambda_2 = \lambda_*\right\}$ and ${\cal R}_1 := \{t \Lambda: 0 < t < 1\ {\rm and}\ \Lambda \in {\bf \Lambda_1}\}$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.1
  • Theorem 1.6
  • Remark 1.2
  • ...and 1 more