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The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds. (English) Zbl 1161.58310
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.

58J05Elliptic equations on manifolds, general theory
58E30Variational principles on infinite-dimensional spaces
Full Text: DOI
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