On the eigenvalues of an elliptic operator $a\left(x,H\left(u\right)\right)$

Campanato, Sergio. "On the eigenvalues of an elliptic operator \( a(x,H(u)) \)." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 3.2 (1992): 107-110. <http://eudml.org/doc/244325>.

@article{Campanato1992,
abstract = {Let \( \Omega \) be a bounded open convex set of class \( C^\{2\} \). Let \( a(x,H(u)) \) be a non linear operator satisfying the condition (A) (elliptic) with constants \( \alpha \), \( \gamma \), \( \delta \). We prove that a number \( \lambda \ge 0 \) is an eigenvalue for the operator \( a(x,H(u)) \) if and only if the number \( \alpha \lambda \) is an eigen-value for the operator \( \Delta u \). If \( \lambda \ge 0 \) , the two systems \( a(x,H(u)) = \lambda u \) and \( \Delta u = \alpha \lambda u \) have the same solutions. In particular, also the eventual eigen-values of the operator \( a(x,H(u)) \) should all be negative. Finally, we obtain a sufficient condition for the existence of solutions \( u \in H^\{2\} \cap H\_\{0\}^\{1\} (\Omega) \) of the system \( a(x,H(u)) = b(x,u,Du) \) where \( b(x,u,p) \) is a vector in \( \mathbb\{R\}^\{N\} \) with a controlled growth.},
author = {Campanato, Sergio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Non linear elliptic systems; Eigen-values; Conditions for existence; eigenvalues of the Laplacian; Carathéodory function},
language = {eng},
month = {6},
number = {2},
pages = {107-110},
publisher = {Accademia Nazionale dei Lincei},
title = {On the eigenvalues of an elliptic operator \( a(x,H(u)) \)},
url = {http://eudml.org/doc/244325},
volume = {3},
year = {1992},
}

TY - JOUR
AU - Campanato, Sergio
TI - On the eigenvalues of an elliptic operator \( a(x,H(u)) \)
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1992/6//
PB - Accademia Nazionale dei Lincei
VL - 3
IS - 2
SP - 107
EP - 110
AB - Let \( \Omega \) be a bounded open convex set of class \( C^{2} \). Let \( a(x,H(u)) \) be a non linear operator satisfying the condition (A) (elliptic) with constants \( \alpha \), \( \gamma \), \( \delta \). We prove that a number \( \lambda \ge 0 \) is an eigenvalue for the operator \( a(x,H(u)) \) if and only if the number \( \alpha \lambda \) is an eigen-value for the operator \( \Delta u \). If \( \lambda \ge 0 \) , the two systems \( a(x,H(u)) = \lambda u \) and \( \Delta u = \alpha \lambda u \) have the same solutions. In particular, also the eventual eigen-values of the operator \( a(x,H(u)) \) should all be negative. Finally, we obtain a sufficient condition for the existence of solutions \( u \in H^{2} \cap H_{0}^{1} (\Omega) \) of the system \( a(x,H(u)) = b(x,u,Du) \) where \( b(x,u,p) \) is a vector in \( \mathbb{R}^{N} \) with a controlled growth.
LA - eng
KW - Non linear elliptic systems; Eigen-values; Conditions for existence; eigenvalues of the Laplacian; Carathéodory function
UR - http://eudml.org/doc/244325
ER -