Let $(M, g)$ be a smooth compact riemannian manifold of dimension $N≥2$ with constant scalar curvature. We are concerned with the following elliptic problem
$\begin{eqnarray*}-{\varepsilon}^2Δ_g u+ u = u^{p-1}, ~~~~u>0,\ \ \ \ \ in \ M.\end{eqnarray*}$
where $Δ_g$ is the Laplace-Beltrami operator on $M$ , $p>2$ if $N = 2$ and $2<p<\frac{2N}{N-2}$ if $N≥3$ , $\varepsilon$ is a small real parameter. We prove that there exist a function $Ξ$ such that if $ξ_0$ is a stable critical point of $Ξ(ξ)$ there exists ${\varepsilon}_0>0$ such that for any ${\varepsilon}∈(0,{\varepsilon}_0)$ , problem (1) has a solution $u_{\varepsilon}$ which concentrates near $ξ_0$ as ${\varepsilon}$ tends to zero. This result generalizes previous works which handle the case where the scalar curvature function of $(M,g)$ has non-degenerate critical points.
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