On a class of GJMS equations on the standard n-sphere

In a recent work by F. Hang and P. Yang (Int. Math. Res. Not. IMRN, 2020), a linear perturbation approach was used to tackle the sharp Sobolev inequality for the Paneitz operator on the standard 3-sphere S^3. This route leads to a class of linearly perturbed critical Paneitz equations of order 4 on S^3. From the geometric point of view, the nonlinearity of these fourth order equations contains a negative exponent which costs some analytical difficulty. As a result and supported by much evidence, it was conjectured by them if positive smooth solutions to these fourth order equations are nothing but constant if the perturbation is sufficiently small in an appropriate sense. In this talk, based on joint work with Ali Hyder, I will introduce new ingredients to tackle the above conjecture in a more general framework. More precisely, we are interested in linearly perturbed critical/subcritical GJMS equations of order 2m on S^n with n<2m. The main finding indicates that if the perturbation is sufficiently small, then any positive smooth solution to these higher-order equations must be constant. In particular, this provides an affirmative answer to the conjecture. Comments on the case n>2m are also made if time permits.