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

Abstract:

Let $D^{1,2} (\mathbb {R}^N)$ denote the closure of $C_0^\infty (\mathbb {R}^N)$ in the norm $\|u\|_{D^{1,2} (\mathbb {R}^N)}^2 = \int \limits _{\mathbb {R}^N} |\nabla u|^2.$ 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 $\varepsilon \geq 0:$ \begin{equation} \tag {$P_\varepsilon $} \left \{\begin {array}{llll} \mbox { Find } u \in D^{1, 2} (\mathbb {R}^N) \mbox { solving}: \\ \left . \begin {array}{lll}-\Delta u &=& \alpha _N (1+ \varepsilon K (x)) u^{\frac {N+2}{N-2}},\\ u &>& 0 \end{array}\right \} \mbox { in } \mathbb {R}^N. \end{array} \right . \end{equation} We show an exact multiplicity result for $(P_\varepsilon )$ for all small $\varepsilon >0$.