On the uniqueness and simplicity of the principal eigenvalue

Marcello Lucia

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2005)

  • Volume: 16, Issue: 2, page 133-142
  • ISSN: 1120-6330

Abstract

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Given an open set Ω of R N N > 2 , bounded or unbounded, and a function w ∈ L N 2 ⁢ Ω with w + ≠ 0 but allowed to change sign, we give a short proof that the positive principal eigenvalue of the problem - △ ⁢ u = λ ⁢ w ⁢ x ⁢ u , u ∈ D 0 1 , 2 ⁢ Ω is unique and simple. We apply this result to study unbounded Palais-Smale sequences as well as to give a priori estimates on the set of critical points of functionals of the type I ⁢ u = 1 2 ⁢ ∫ Ω ∇ ⁡ u 2 ⁢ d ⁢ x - ∫ Ω G ⁢ x , u ⁢ d ⁢ x , u ∈ D 0 1 , 2 ⁢ Ω , when G has a subquadratic growth at infinity.

How to cite

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Lucia, Marcello. "On the uniqueness and simplicity of the principal eigenvalue." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 16.2 (2005): 133-142. <http://eudml.org/doc/252327>.

@article{Lucia2005,
abstract = {Given an open set $\Omega$ of $\mathbb\{R\}^\{N\}$$(N > 2)$, bounded or unbounded, and a function $w \in L^\{\frac\{N\}\{2\}\} (\Omega)$ with $w^\{+\}\neq 0$ but allowed to change sign, we give a short proof that the positive principal eigenvalue of the problem $$ - \triangle u = \lambda w (x) u, \qquad u \in \mathcal\{D\}^\{1,2\}\_\{0\} (\Omega)$$ is unique and simple. We apply this result to study unbounded Palais-Smale sequences as well as to give a priori estimates on the set of critical points of functionals of the type $$I(u) = \frac\{1\}\{2\}\int\_\{\Omega\} |\nabla u|^\{2\} \, dx - \int\_\{\Omega\} G(x,u) \, dx, \quad u \in \mathcal\{D\}^\{1,2\}\_\{0\} (\Omega),$$ when $G$ has a subquadratic growth at infinity.},
author = {Lucia, Marcello},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Principal eigenvalue; Simple eigenvalue; Capacity; Palais Smale sequence; principal eigenvalue; simple eigenvalue; capacity; Palais-Smale sequence},
language = {eng},
month = {6},
number = {2},
pages = {133-142},
publisher = {Accademia Nazionale dei Lincei},
title = {On the uniqueness and simplicity of the principal eigenvalue},
url = {http://eudml.org/doc/252327},
volume = {16},
year = {2005},
}

TY - JOUR
AU - Lucia, Marcello
TI - On the uniqueness and simplicity of the principal eigenvalue
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2005/6//
PB - Accademia Nazionale dei Lincei
VL - 16
IS - 2
SP - 133
EP - 142
AB - Given an open set $\Omega$ of $\mathbb{R}^{N}$$(N > 2)$, bounded or unbounded, and a function $w \in L^{\frac{N}{2}} (\Omega)$ with $w^{+}\neq 0$ but allowed to change sign, we give a short proof that the positive principal eigenvalue of the problem $$ - \triangle u = \lambda w (x) u, \qquad u \in \mathcal{D}^{1,2}_{0} (\Omega)$$ is unique and simple. We apply this result to study unbounded Palais-Smale sequences as well as to give a priori estimates on the set of critical points of functionals of the type $$I(u) = \frac{1}{2}\int_{\Omega} |\nabla u|^{2} \, dx - \int_{\Omega} G(x,u) \, dx, \quad u \in \mathcal{D}^{1,2}_{0} (\Omega),$$ when $G$ has a subquadratic growth at infinity.
LA - eng
KW - Principal eigenvalue; Simple eigenvalue; Capacity; Palais Smale sequence; principal eigenvalue; simple eigenvalue; capacity; Palais-Smale sequence
UR - http://eudml.org/doc/252327
ER -

References

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