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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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