A prescribed scalar curvature-type equation: almost critical manifolds and multiple solutions

We present an asymptotic analysis for a perturbed prescribed scalar curvature-type equation. A major consequence is a non-existence result in low dimension. Conversely, we prove an existence result in higher dimensions: to this aim we develop a general finite-dimensional reduction procedure for perturbed variational functionals. The general principle can be useful to discuss some other nonlinear elliptic PDE with Sobolev critical growth in bounded domains