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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 $$ -\varepsilon^2\Delta_g u +u=u^{p-1}\ \text{in}\ M,\quad u > 0 \ \text{in}\ M $$ has a $K$-peaks solution. Here $p > 2$ if $N = 2$ and $2 < p < 2^*=\frac{2N}{N-2}$ when $N\geq 3.$ Moreover, the peaks collapse, as $\varepsilon\to0,$ to an isolated local minimum point of the scalar curvature.

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