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Asymptotic behaviour of ground states. (English) Zbl 0860.35029
The authors study ground state solutions of the following system of semilinear elliptic equations in $$\mathbb{R}^n$$ $$(n>2)$$: $-\Delta u=|v |^{q-1} v,\quad - \Delta v= |u |^{p-1}u, \tag{*}$ i.e. solutions which are minimal critical points of the functional $J_{p,q} (u,v) =\int_{\mathbb{R}^n} \nabla u \nabla v- {1\over p+1} \int_{\mathbb{R}^n} |u |^{p+1} - {1\over q+ 1} \int_{\mathbb{R}^n} |v |^{q+1}.$ Under the condition $$n/(p+1) + n/(q+1) = n-2$$, they prove the existence of a unique up to translations and scalings), positive, radially symmetric ground state of (*). In addition they study the asymptotic behaviour, as $$|x|\to\infty$$, of this solution.

##### MSC:
 35J60 Nonlinear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs 35J50 Variational methods for elliptic systems 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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