Exact multiplicity result for the perturbed scalar curvature problem in $\mathbb{R}^N$ $(N \geq 3)$

Let $D^{1,2} (\mathbb{R}^N)$ denote the closure of $C_0^\infty (\mathbb{R}^N)$ in the norm $\Vert u\Vert _{D^{1,2} (\mathbb{R}^N)}^2 = \int\limits_{\mathbb{R}^N} \vert\nabla u\vert^2.$ Let $N \geq 3$ and define the constants $\alpha_N = N (N-2)$ and $C_N = (N (N-2))^{\frac{N-2}{4}}.$ Let $K \in C^2 (\mathbb{R}^N).$ We consider the following problem for $\varepsilon \geq 0:$

$(P_\varepsilon)\qquad\qquad\quad \left\{\begin{array}{llll} \mbox... ...array}\right\} \mbox{ in } \mathbb{R}^N. \end{array} \right. \qquad\quad\qquad$

We show an exact multiplicity result for $(P_\varepsilon)$ for all small $\varepsilon >0$.