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Two solutions for a fourth order nonlocal problem with indefinite potentials

  • Giovany M. Figueiredo
  • Marcelo F. Furtado
  • João Pablo P. da Silva
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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.

Mathematics Subject Classification

Primary 35J60 Secondary 35J20 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Giovany M. Figueiredo
    • 1
  • Marcelo F. Furtado
    • 1
  • João Pablo P. da Silva
    • 2
  1. 1.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal do ParáSoureBrazil

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