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Perturbation from symmetry for indefinite semilinear elliptic equations

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Abstract

We prove the existence of an unbounded sequence of solutions for an elliptic equation of the form \({-\Delta u=\lambda u + a(x)g(u)+f(x), u\in H^1_0(\Omega)}\), where \({\lambda \in \mathbb{R}, g(\cdot)}\) is subcritical and superlinear at infinity, and a(x) changes sign in Ω; moreover, g( − s) =  − g(s) \({\forall s}\). The proof uses Rabinowitz’s perturbation method applied to a suitably truncated problem; subsequent energy and Morse index estimates allow us to recover the original problem. We consider the case of \({\Omega\subset \mathbb{R}^N}\) bounded as well as \({\Omega=\mathbb{R}^N, \, N\geqslant 3}\).

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Authors and Affiliations

  1. CMAF, Faculty of Science, University of Lisbon, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal

    Miguel Ramos

  2. Department of Mathematical Sciences, University of Nevada, Las Vegas, NV, 89154-4020, USA

    Hossein Tehrani

Authors
  1. Miguel Ramos
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  2. Hossein Tehrani
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Correspondence to Miguel Ramos.

Additional information

M. Ramos’s research was supported by Fundação para a Ciência e a Tecnologia, Financiamento Base 2008—ISFL/1/209.

This work was completed while H. Tehrani was visiting IST Lisbon on a sabbatical from UNLV. The support of both institutions is gratefully acknowledged.

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Ramos, M., Tehrani, H. Perturbation from symmetry for indefinite semilinear elliptic equations. manuscripta math. 128, 297–314 (2009). https://doi.org/10.1007/s00229-008-0228-1

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  • DOI: https://doi.org/10.1007/s00229-008-0228-1

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