On the existence of a component-wise positive radially symmetric solution for a superlinear system

P. Zhidkov, Bogoliubov Laboratory of Theoretical Physics, Dubna, Russia

E. J. Qualitative Theory of Diff. Equ., No. 29. (2007), pp. 1-7.

Abstract: The system under consideration is
$$
-\Delta u+a_uu=u^3-\beta uv^2, \quad u=u(x),
$$
$$
-\Delta v+a_vv=v^3-\beta u^2v, \quad v=v(x), \ x\in \mathbb {R}^3,
$$
$$
u\big| _{|x|\to \infty }=v\big| _{|x|\to \infty }=0,
$$
where $a_u,a_v$ and $\beta $ are positive constants. We prove the existence of a component-wise positive smooth radially symmetric solution of this system. This result is a part of the results presented in the recent paper by Sirakov [1]; in our opinion, our method allows one to treat the problem simpler and shorter.

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