Stationary Kirchhoff systems in closed high dimensional manifolds (Q2796990)

In the interesting paper under review, the authors discuss existence, compactness and stability properties of solutions to a Kirchhoff-type systems NEWLINE\[NEWLINE \left( a+b\sum_{j=1}^p \int_M | \nabla u_j|^2 dv_g\right) \Delta_g u_i + \sum_{j=1}^p A_{ij}u_j=| U|^{2^*-2}u_i,\quad u_i\geq 0,\quad i=1,\dots,pNEWLINE\]NEWLINE over a closed \(n\)-manifolds \((M^n,g)\), \(n\geq 4\). Here \(\Delta_g\) is the Laplace-Beltrami operator, \(A=\{A_{ij}\}\) is a \(C^1\)-map from \(M\) into the space of the symmetric \(p\times p\) matrices with real entries, \(U=(u_1,\dots,u_p),\) \(| U|:\;M\to \mathbb R\) is the Euclidean norm of \(U\) and \(2^*=\frac{2n}{n-2}\) is the critical Sobolev exponent.