Existence of positive solutions of the Lane-Emden system. (English) Zbl 0917.35031

The weakly coupled system \[ \Delta u+ u^p= 0,\quad \Delta v+ v^q= 0\quad\text{in }\mathbb{R}^n,\quad n\geq 3, \] is called the Lane-Emden system. The exponents \(p\), \(q>0\) are critical if \[ 1/(p+ 1)+ 1/(q+1)= (n-2)/n\tag{\(*\)} \] and the authors show that in this case or in the supercritical case, i.e. \((*)\) holds with \(<\), there are infinitely many values \(\xi_i>0\), \(i= 1,2\), such that the Lane-Emden system has a positive radial solution with \(u(0)= \xi_1\), \(v(0)= \xi_2\) and \(u,v\to 0\) as \(| x|\to \infty\). The proof is based on a shooting argument for an ordinary differential system associated to radial solutions.


35J45 Systems of elliptic equations, general (MSC2000)
35J60 Nonlinear elliptic equations