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Abstract In this paper, we study the problem of prescribing Q -Curvature on higher dimensional standard spheres. The problem consists in finding the right assumptions on a function K so that it is the Q -Curvature of a metric conformal to the standard one on the sphere. Using some pinching condition, we track the change in topology that occurs when crossing a critical level (or a virtually critical level if it is a critical point at infinity) and then compute a certain Euler-Poincaré index which allows us to prove the existence of many solutions. The locations of the levels sets of these solutions are determined in a very precise manner. These type of multiplicity results are new and are proved without any assumption of symmetry or periodicity on the function K .
Ayed et al. (Sat,) studied this question.