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

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

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

$-{\epsilon }^{2}{{\Delta }}_{g}u+u={u}^{p-1}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}M,\phantom{\rule{1.em}{0ex}}u>0\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}M$

has a $K$-peaks solution. Here $p>2$ if $N=2$ and $2 when $N\ge 3·$ Moreover, the peaks collapse, as $\epsilon \to 0,$ to an isolated local minimum point of the scalar curvature.

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