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Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifolds. (English) Zbl 1159.58012

Let (M,g) be a smooth, compact, N-dimensional Riemannian manifold.

The authors prove that for any fixed positive integer K the problem

-ε 2 Δ g u+u=u p-1 inM,u>0inM

has a K-peaks solution. Here p>2 if N=2 and 2<p<2 * =2N N-2 when N3· Moreover, the peaks collapse, as ε0, to an isolated local minimum point of the scalar curvature.

MSC:
58J05Elliptic equations on manifolds, general theory
58E30Variational principles on infinite-dimensional spaces
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