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Singularly Perturbed Elliptic Equations with Symmetry: Existence of Solutions Concentrating on Spheres, Part I

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Abstract:

 We deal with the existence of positive radial solutions concentrating on spheres to a class of singularly perturbed elliptic problems like −ɛ2Δu+V(|x|)u=u p,uH 1(ℝn). Under suitable assumptions on the auxiliary potential M(r)=r n−1 V θ(r), θ(p+1)/(p−1)−1/2, we provide necessary and sufficient conditions for concentration as well as the bifurcation of non-radial solutions.

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Authors and Affiliations

  1. S.I.S.S.A., 2-4 via Beirut, 34014 Trieste, Italy. E-mail: ambr@sissa.it, , , , , , IT

    Antonio Ambrosetti

  2. School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540 USA. E-mail: malchiod@ias.edu, , , , , , US

    Andrea Malchiodi

  3. School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA. E-mail: ni@math.umn.edu, , , , , , US

    Wei-Ming Ni

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  1. Antonio Ambrosetti
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  2. Andrea Malchiodi
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  3. Wei-Ming Ni
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Received: 20 September 2002 / Accepted: 28 October 2002 Published online: 28 February 2003

Communicated by P. Constantin

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Ambrosetti, A., Malchiodi, A. & Ni, WM. Singularly Perturbed Elliptic Equations with Symmetry: Existence of Solutions Concentrating on Spheres, Part I. Commun. Math. Phys. 235, 427–466 (2003). https://doi.org/10.1007/s00220-003-0811-y

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  • DOI: https://doi.org/10.1007/s00220-003-0811-y

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