Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifolds期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:

Institution:

(1) School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia;(2) Dipartimento di Matematica Applicata, “U.Dini”, Università di Pisa, via F. Buonarroti 1/c, 56100 Pisa, Italy;(3) Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, via A. Scarpa 16, 00161 Rome, Italy

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 ${2 < p < 2^*={2N \over N-2}\,if\,N\ge3}$ . 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”.

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 58G03 58E30

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