Electron. J. Diff. Equ., Vol. 2013 (2013), No. 82, pp. 1-11.

Positive solutions and global bifurcation of strongly coupled elliptic systems

Jagmohan Tyagi

Abstract:
In this article, we study the existence of positive solutions for the coupled elliptic system
$$\displaylines{ -\Delta u= \lambda (f(u, v)+ h_{1}(x) ) \quad \hbox{in }\Omega, \cr -\Delta v= \lambda (g(u, v)+ h_{2}(x))\quad \hbox{in }\Omega, \cr u =v=0 \quad \hbox{on }\partial \Omega, }$$
under certain conditions on $f,g$ and allowing $h_1, h_2$ to be singular. We also consider the system
$$\displaylines{ -\Delta u= \lambda ( a(x) u + b(x)v+ f_{1}(v)+ f_{2}(u) ) \quad \hbox{in } \Omega, \cr -\Delta v= \lambda ( b(x)u+ c(x)v+ g_{1}(u)+ g_{2}(v) )\quad \hbox{in }\Omega , \cr u =v=0 \quad \hbox{on }\partial \Omega, }$$
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.

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Jagmohan Tyagi
Indian Institute of Technology Gandhinagar
Vishwakarma Government Engineering College Complex
Chandkheda, Visat-Gandhinagar Highway, Ahmedabad
Gujarat, India, 382424
email: jtyagi1@gmail.com, jtyagi@iitgn.ac.in

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