Existence of critical points of a conformally invariant variational problem (Q2779207)

The author considers the existence of critical points of the functional NEWLINE\[NEWLINEE(u)=\int_M \bigl(\kappa |\nabla u|^2+ R_gu^2\bigr) dV_g,\quad \kappa= {4(n-1) \over n-2},NEWLINE\]NEWLINE on the constrained set NEWLINE\[NEWLINE{\mathcal C}(M)= \left \{u\in H^1(M): \int_M|u|^N dV_g=1\right\}, \quad N={2n\over n-2},NEWLINE\]NEWLINE where \((M,g)\) is a compact Riemannian manifold of dimension \(n\geq 3\), \(R_g\) is the scalar curvature and \(dV_g\) is the volume element of \((M,g)\).NEWLINENEWLINENEWLINEThe paper outlines some results in the field. A first theorem describes the behaviour of minimizing sequences. A next theorem gives the existence of infinitely many critical points, in case when \(M\) admits an action of a compact Lie group. Finally some applications to partial differential equations in the Euclidean space are indicated.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00018].