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FitzHugh-Nagumo system with zero mass and critical growth

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Abstract

We show existence of a nontrivial nonnegative solution for the system \( - \Delta u = K(x)f(u) + \gamma {\left| u \right|^{{2^ *} - 2}}u - v,\, - \Delta v = u - v\) in ℝN. Since the function f can verify \(f\prime (0) = 0\), this type of system is known in the literature as zero mass. We analyze three types of problems with K being periodic, asymptotically periodic and with a vanishing property at infinity. In the first place we consider N ≥ 3, and we prove existence results considering the function f with polynomial growth which can be subcritical, corresponding to γ = 0, or critical, in case γ = 1. Finally, we consider specifically N = 2 with γ = 0 and f with possible critical exponential behavior.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade de Brasília, Campus Darcy Ribeiro 01, Brasília, DF, CEP 70910-900, Brasil

    Giovany Figueiredo

  2. IMECC, Departamento de Matemática, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda 651, Campinas, SP, CEP 13083-859, Brasil

    Marcelo Montenegro

Authors
  1. Giovany Figueiredo
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  2. Marcelo Montenegro
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Correspondence to Marcelo Montenegro.

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Supported in part by CNPQ, FAPDF and CAPES.

Supported in part by CNPq and FAPESP.

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Figueiredo, G., Montenegro, M. FitzHugh-Nagumo system with zero mass and critical growth. Isr. J. Math. 245, 711–733 (2021). https://doi.org/10.1007/s11856-021-2224-z

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  • DOI: https://doi.org/10.1007/s11856-021-2224-z

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