Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 82, pp. 1-11.
Title: Positive solutions and global bifurcation of strongly coupled elliptic systems
Author: Jagmohan Tyagi (Vishwakarma Government Engineering College, India)
Abstract:
In this article, we study the existence of positive solutions for
the coupled elliptic system
\begin{gather*}
-\Delta u= \lambda (f(u, v)+ h_{1}(x) ) \quad \text{in }\Omega, \\
-\Delta v= \lambda (g(u, v)+ h_{2}(x))\quad \text{in }\Omega, \\
u =v=0 \quad \text{on }\partial \Omega,
\end{gather*}
under certain conditions on $f,g$ and allowing $h_1, h_2$ to be singular.
We also consider the system
\begin{gather*}
-\Delta u= \lambda ( a(x) u + b(x)v+ f_{1}(v)+ f_{2}(u) ) \quad \text{in }\Ome
ga, \\
-\Delta v= \lambda ( b(x)u+ c(x)v+ g_{1}(u)+ g_{2}(v) )\quad \text{in }\Omega
, \\
u =v=0 \quad \text{on }\partial \Omega,
\end{gather*}
and prove a Rabinowitz global bifurcation type theorem to this system.
Submitted April 18, 2012. Published March 31, 2013.
Math Subject Classifications: 35J57, 35B32, 35B09.
Key Words: Elliptic system; bifurcation; positive solutions.