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Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifolds

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Abstract

Given (M, g) a smooth compact Riemannian N-manifold, we prove that for any fixed positive integer K the problem

$$-\varepsilon^2\Delta_g u +u=u^{p-1}\,{{\rm in}\, M}, \quad u > 0\,{{\rm in}\,M}$$

has a K-peaks solution, whose peaks collapse, as ε goes to zero, to an isolated local minimum point of the scalar curvature. Here p > 2 if N = 2 and \({2 < p < 2^*={2N \over N-2}\,if\,N\ge3}\).

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Correspondence to E. N. Dancer.

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E. N. Dancer was partially supported by the ARC. A. M. Micheletti and A. Pistoia are supported by Mi.U.R. Project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.

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Dancer, E.N., Micheletti, A.M. & Pistoia, A. Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifolds. manuscripta math. 128, 163–193 (2009). https://doi.org/10.1007/s00229-008-0225-4

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  • DOI: https://doi.org/10.1007/s00229-008-0225-4

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