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A Critical Concave–Convex Kirchhoff-Type Equation in \(\mathbb R^4\) Involving Potentials Which May Vanish at Infinity

Abstract

We establish the existence and multiplicity of solutions for a Kirchhoff-type problem in \(\mathbb R^4\) involving a critical and concave–convex nonlinearity. Since in dimension four, the Sobolev critical exponent is \(2^*=4\), there is a tie between the growth of the nonlocal term and the critical nonlinearity. This turns out to be a challenge to study our problem from the variational point of view. Some of the main tools used in this paper are the mountain-pass and Ekeland’s theorems, Lions’ Concentration Compactness Principle and an extension to \(\mathbb R^N\) of the Struwe’s global compactness theorem.

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Notes

  1. 1.

    \(K\equiv 1\) in the whole \(\mathbb {R}^4\) in second case.

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Acknowledgements

The authors would like to express their gratitude to the anonymous referees for their carefully reading of the manuscript with valuable comments and suggestions.

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Correspondence to Marcelo C. Ferreira.

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Supported by Fondecyt 1181125.

Communicated by Nader Masmoudi.

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Ferreira, M.C., Ubilla, P. A Critical Concave–Convex Kirchhoff-Type Equation in \(\mathbb R^4\) Involving Potentials Which May Vanish at Infinity. Ann. Henri Poincaré (2021). https://doi.org/10.1007/s00023-021-01105-5

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Mathematics Subject Classification

  • 35B33
  • 35J20
  • 35J60