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Zbl 1161.58310
Micheletti, Anna Maria; Pistoia, Angela
The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds. (English)
[J] Calc. Var. Partial Differ. Equ. 34, No. 2, 233-265 (2009). ISSN 0944-2669; ISSN 1432-0835/e

Let $(M, g)$ be a smooth compact Riemannian $N$-manifold, $N \geq 2,$ and $p > 2$ if $N = 2$ and ${2 < p < 2^{*} = {2N \over N-2}}$ if $N \geq 3.$ The authors show that the positive solutions of the problem $$ -\varepsilon^2\Delta_g u + u = u^{p-1}\quad \text{in}\ M $$ are generated by stable critical points of the scalar curvature of $g,$ if $\varepsilon$ is small enough.
[Dian K. Palagachev (Bari)]

MSC 2000:: *58J05 Elliptic equations on manifolds, general theory
58E30 Variational principles on infinite-dimensional spaces

Keywords: Riemannian manifold; scalar curvature; nonlinear elliptic equation

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