On positive solutions of quasilinear elliptic systems

Yuanji Cheng

Czechoslovak Mathematical Journal (1997)

  • Volume: 47, Issue: 4, page 681-687
  • ISSN: 0011-4642

Abstract

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In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems - Δ p u = f ( x , u , v ) , in Ω , - Δ p v = g ( x , u , v ) , in Ω , u = v = 0 , on ∂ Ω , where - Δ p is the p -Laplace operator, p > 1 and Ω is a C 1 , α -domain in ℝ n . We prove an analogue of [7, 16] for the eigenvalue problem with f ( x , u , v ) = λ 1 v p - 1 , g ( x , u , v ) = λ 2 u p - 1 and obtain a non-existence result of positive solutions for the general systems.

How to cite

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Cheng, Yuanji. "On positive solutions of quasilinear elliptic systems." Czechoslovak Mathematical Journal 47.4 (1997): 681-687. <http://eudml.org/doc/30391>.

@article{Cheng1997,
abstract = {In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems \[ \left\rbrace \begin\{array\}\{ll\}-\Delta \_p u = f(x,u,v), &\quad \text\{in\} \ \Omega , -\Delta \_p v = g(x,u,v), &\quad \text\{in\} \ \Omega , u = v = 0, &\quad \text\{on\} \ \partial \Omega , \end\{array\}\right.\] where $-\Delta _p$ is the $p$-Laplace operator, $p>1$ and $\Omega $ is a $C^\{1,\alpha \}$-domain in $\mathbb \{R\}^n$. We prove an analogue of [7, 16] for the eigenvalue problem with $f(x,u,v)=\lambda _1 v^\{p-1\}$, $ g(x,u,v)=\lambda _2u^\{p-1\}$ and obtain a non-existence result of positive solutions for the general systems.},
author = {Cheng, Yuanji},
journal = {Czechoslovak Mathematical Journal},
keywords = {Eigenvalue problem; Degenerate elliptic operator; Nonlinear systems; Positive solutions; eigenvalue problem; -Laplacian},
language = {eng},
number = {4},
pages = {681-687},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On positive solutions of quasilinear elliptic systems},
url = {http://eudml.org/doc/30391},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Cheng, Yuanji
TI - On positive solutions of quasilinear elliptic systems
JO - Czechoslovak Mathematical Journal
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 47
IS - 4
SP - 681
EP - 687
AB - In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems \[ \left\rbrace \begin{array}{ll}-\Delta _p u = f(x,u,v), &\quad \text{in} \ \Omega , -\Delta _p v = g(x,u,v), &\quad \text{in} \ \Omega , u = v = 0, &\quad \text{on} \ \partial \Omega , \end{array}\right.\] where $-\Delta _p$ is the $p$-Laplace operator, $p>1$ and $\Omega $ is a $C^{1,\alpha }$-domain in $\mathbb {R}^n$. We prove an analogue of [7, 16] for the eigenvalue problem with $f(x,u,v)=\lambda _1 v^{p-1}$, $ g(x,u,v)=\lambda _2u^{p-1}$ and obtain a non-existence result of positive solutions for the general systems.
LA - eng
KW - Eigenvalue problem; Degenerate elliptic operator; Nonlinear systems; Positive solutions; eigenvalue problem; -Laplacian
UR - http://eudml.org/doc/30391
ER -

References

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