Nonlinear elliptic equations involving critical Sobolev exponent on compact Riemannian manifolds in presence of symmetries (Q1300572)

Let \((M , g)\) be a compact, smooth, Riemannian \(n\)-manifold of dimension greater than 3, consider \( q \in (1; (n+2)/(n-2))\) and let \(a , f \) and \(h\) be three smooth functions on \(M\). The author studies smooth, positive solutions \(u\) of the equation \( \Delta _{g} u+au= f u^{ (n+2)/(n-2) }+ h u ^{ q }\), assuming that a subgroup \(G\) of \(\text{Isom}_{g}(M)\) is fixed and that the functions \(a\), \(f\), \(h\) are \(G \)-invariant. Under suitable assumptions which relate the orbits of \(G\) to \(f\), \(a\) and \(h\), the authors show that the equation admits \(G\)-invariant solutions.