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Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems

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Abstract

We study the existence and asymptotic behavior of positive and sign-changing multipeak solutions for the equation

$$ -\varepsilon^2\Delta v+V(x)v=f(v)\quad{\rm in}\,\,\,\mathbb{R}^N, $$

where ε is a small positive parameter, f a superlinear, subcritical and odd nonlinearity, V a uniformly positive potential. No symmetry on V is assumed. It is known (Kang and Wei in Adv Differ Equ 5:899–928, 2000) that this equation has positive multipeak solutions with all peaks approaching a local maximum of V. It is also proved that solutions alternating positive and negative spikes exist in the case of a minimum (see D’Aprile and Pistoia in Ann Inst H. Poincaré Anal Non Linéaire 26:1423–1451, 2009). The aim of this paper is to show the existence of both positive and sign-changing multipeak solutions around a nondegenerate saddle point of V.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133, Rome, Italy

    Teresa D’Aprile

  2. Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Avda. Fuentenueva s/n, 18071, Granada, Spain

    David Ruiz

Authors
  1. Teresa D’Aprile
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  2. David Ruiz
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Correspondence to David Ruiz.

Additional information

T. D’Aprile has been supported by the Italian PRIN Research Project 2007. Metodi variazionali e topologici nello studio di fenomeni non lineari. D. Ruiz has been supported by the Spanish Ministry of Science and Innovation under Grant MTM2008-00988 and by J. Andalucía (FQM 116).

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D’Aprile, T., Ruiz, D. Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems. Math. Z. 268, 605–634 (2011). https://doi.org/10.1007/s00209-010-0686-5

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