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Ground states of elliptic problems over cones

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Abstract

Given a reflexive Banach space X, we consider a class of functionals \(\Phi \in C^1(X,{\mathbb {R}})\) that do not behave in a uniform way, in the sense that the map \(t \mapsto \Phi (tu)\), \(t>0\), does not have a uniform geometry with respect to \(u\in X\). Assuming instead such a uniform behavior within an open cone \(Y \subset X \setminus \{0\}\), we show that \(\Phi \) has a ground state relative to Y. Some further conditions ensure that this relative ground state is the (absolute) ground state of \(\Phi \). Several applications to elliptic equations and systems are given.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade de Brasília, Brasília, DF, 70910-900, Brazil

    Giovany M. Figueiredo

  2. CIEM-FaMAF, Universidad Nacional de Córdoba, 5000, Córdoba, Argentina

    Humberto Ramos Quoirin

  3. Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, GO, 74001-970, Brazil

    Kaye Silva

Authors
  1. Giovany M. Figueiredo
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  2. Humberto Ramos Quoirin
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  3. Kaye Silva
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Correspondence to Giovany M. Figueiredo.

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Communicated by P. H. Rabinowitz.

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The first author was partially supported by CNPq/Brazil under the grants [304657/2018-2] and [407479/2018-0], and FAPDf 04/2017. The third author was partially supported by CNPq/Brazil under the grant [408604/2018-2].

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Figueiredo, G.M., Ramos Quoirin, H. & Silva, K. Ground states of elliptic problems over cones. Calc. Var. 60, 189 (2021). https://doi.org/10.1007/s00526-021-02052-z

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