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Positive solution for an indefinite fourth-order nonlocal problem

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Abstract

We prove the existence of a positive solution for the problem \({\rm{\gamma}}{{\rm{\Delta}}^2}u - m\left(u \right){\rm{\Delta}}u = \mu a\left(x \right){u^q} + b\left(x \right){u^p},\,\,{\rm{in}}\,{\rm{\Omega ,}}\,\,\,\,\,u = {\rm{\gamma \Delta}}u = 0,\,\,{\rm{on}}\,\,\partial {\rm{\Omega ,}}\) where Ω ⊂ ℝN is a bounded smooth domain, γ ∈ {0, 1},0 < q > 1 < p, m is weakly continuous in \({H^2}\left({\rm{\Omega}} \right) \cap H_0^1\left({\rm{\Omega}} \right),a \in {L^\infty}\left({\rm{\Omega}} \right)\) is nonnegative and b is a bounded potential which can change sign. The solution is obtained via a sub-supersolution approach when the parameter µ > 0 is small.

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Author information

Authors and Affiliations

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

    Marcelo F. Furtado

  2. Departamento de Matemática, Universidade Federal do Pará, 66075-110, Belém-PA, Brazil

    João Pablo P. da Silva

Authors
  1. Marcelo F. Furtado
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  2. João Pablo P. da Silva
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Correspondence to Marcelo F. Furtado.

Additional information

The first author was partially supported by CNPq and FAPDF/Brazil.

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Furtado, M.F., da Silva, J.P.P. Positive solution for an indefinite fourth-order nonlocal problem. Isr. J. Math. 241, 775–794 (2021). https://doi.org/10.1007/s11856-021-2104-6

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  • DOI: https://doi.org/10.1007/s11856-021-2104-6

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