MULTIPLE POSITIVE SOLUTIONS OF A SCALAR FIELD EQUATION IN Rn

(see for example [3], [8], [9], [14], [17]–[19], [24] and the references therein). Here, λ ∈ R is a positive parameter, k is a given smooth function on R, n ≥ 3, and 2∗ = 2n/(n− 2) is the critical Sobolev exponent. Problem (0) has a geometrical relevance, since for λ = 0 every solution to (0) gives rise, up to a stereographic projection, to a metric g on the sphere whose scalar curvature is proportional to k(x). From the point of view of the Calculus of Variations the interest in the Kazdan–Warner problem is due to the role of the noncompact group of dilations in R. This produces quite a large spectrum of phenomena, like concentrations of maps, lack of compactness, failure of the Palais–Smale condition and nonexistence results. In the spirit of the paper by Coron [11] (see also [2]) one may ask if the coefficient k(x) affects the topology of the energy sublevels. In this paper we give an answer to this question in the subcritical case. Namely, we study the