Multiple solutions for critical elliptic systems in potential form (Q928323)

Let \((M,g)\) be a smooth and compact \(n\)-dimensional Riemannian manifold with \(n\geq3.\) For an integer \(p\geq1\) set \(M^s_p(\mathbb R)\) for the vector space of symmetric \(p\times p\) real matrices. Given a smooth \(A\colon M\to M^s_p(\mathbb R),\) consider the vector valued equation \[ \Delta^p_g {\mathcal U}+A(x){\mathcal U}={1\over {2^*}} D_{\mathcal U} | {\mathcal U}| ^{2^*} \] where \({\mathcal U}\colon M\to \mathbb R^p\) is a map, \(\Delta^p_g\) is the Laplace--Beltrami operator over \(M\) acting on \(p\)-maps, \(2^*=2n/(n-2)\) and \(D_{\mathcal U}\) stands for the derivation with respect to \({\mathcal U}.\) Setting \({\mathcal U}=(u_1,\ldots,u_p)\) and noting that \(\Delta^p_g{\mathcal U}=(\Delta_g u_i)_i\) with the Laplace--Beltrami operator \(\Delta_g\) acting on functions, the above equation could be written as the elliptic system \[ \Delta_g u_i +\sum_{j=1}^p A_{ij}(x) u_j=| u_i| ^{2^*-2}u_i,\quad i=1,\ldots,p. \] By virtue of topological arguments based on the Lusternik--Schnirelmann category, the authors study multiplicity of solutions to that system.