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.
Esposito, P., Mancini, G. (2003). A prescribed scalar curvature-type equation: almost critical manifolds and multiple solutions. JOURNAL OF FUNCTIONAL ANALYSIS, 205(2), 306-356 [10.1016/j.jfa.2003.07.004].