From the viewpoint of the calculus of variations, the perturbed Kazdan-Warner problem:
(1) −∆u+λu=k(x)u^{p−1}, u>0 in R^n, u→0 at ∞,
where n≥3 and p>1 is subcritical. Problem (1) is studied with regard of the effect of the set M on topology of the energy sub levels: in the main results it is shown that the Lyusternik-Schnirelman category of M can
affect the number of positive solutions to (1) in case p is close enough to the critical Sobolev exponent.
