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Bifurcations of radially symmetric solutions in a coupled elliptic system with critical growth in \(\mathbb{R}^d\) for \(d=3,4\). (English) Zbl 1435.34029

In the paper under review the authors consider the coupled elliptic partial differential equations \[ \begin{aligned} &\Delta u + (|u|^{p-1} + \beta_1 |v|^{p-1}) u =0,\\ &\Delta v + (\beta_1 |u|^{p-1} + \beta_2 |v|^{p-1}) v =0 \end{aligned} \] in the entire space \(\mathbb{R}^{d}\) with \(d\geq3\) where \((u,v)=(u(x),v(x))\) are real-valued functions on \(\mathbb{R}^{d}\), \(\Delta\) is the Laplace operator on \(\mathbb{R}^{d}\) and \(p, \beta_1, \beta_2 \in\mathbb{R}\) are constants such that \(p=(d+2)/(d-2)\) and \(\beta_2>0\). They investigate the bifurcations of three families of radially symmetric, bounded solutions. To this end, first they reduce the problems of the three families radially symmetric homoclinic orbits in a four-dimensional reversible system of ordinary differential equations. It is shown that transcritical or pitchfork bifurcations of the three families occur at infinitely many parameter values. Finally, numerical computations for symmetric homoclinic orbits and radially symmetric bounded solutions are given.

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
35J57 Boundary value problems for second-order elliptic systems
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37M20 Computational methods for bifurcation problems in dynamical systems

Software:

AUTO-07P; AUTO; HomCont
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References:

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