## 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.

### MSC:

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