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Two solutions for a fourth order nonlocal problem with indefinite potentials

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Abstract

We study the nonlocal equation

$$\begin{aligned} \Delta ^{2}u-m\left( \displaystyle \int _{\Omega }|\nabla u|^{2} dx \right) \Delta u = \lambda a(x) |u|^{q-2}u+ b(x)|u|^{p-2}u, \, \text{ in } \Omega , \end{aligned}$$

subject to the boundary condition \(u=\Delta u=0\) on \(\partial \Omega \). For m continuous and positive we obtain a nonnegative solution if \(1<q<2<p \le 2N/(N-4)\) and \(\lambda >0\) small. If the affine case \(m(t)=\alpha +\beta t\), we obtain a second solution if \(4<p<2N/(N-4)\) and \(N \in \{5,6,7\}\). In the proofs we apply variational methods.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade de Brasília, Brasília, Brazil

    Giovany M. Figueiredo & Marcelo F. Furtado

  2. Departamento de Matemática, Universidade Federal do Pará, Soure, Brazil

    João Pablo P. da Silva

Authors
  1. Giovany M. Figueiredo
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  2. Marcelo F. Furtado
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  3. João Pablo P. da Silva
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Correspondence to Giovany M. Figueiredo.

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Figueiredo, G.M., Furtado, M.F. & da Silva, J.P.P. Two solutions for a fourth order nonlocal problem with indefinite potentials. manuscripta math. 160, 199–215 (2019). https://doi.org/10.1007/s00229-018-1057-5

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  • DOI: https://doi.org/10.1007/s00229-018-1057-5

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