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A Nonhomogeneous and Critical Kirchhoff–Schrödinger Type Equation in \(\mathbb R^4\) Involving Vanishing Potentials

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Abstract

We study the existence and multiplicity of weak solutions for a Kirchhoff–Schrödinger type problem in \(\mathbb R^4\) involving a critical nonlinearity and a suitable small perturbation. When \(N=4\), the Sobolev exponent is \(2^*=4\) and, as a consequence, there is a tie between the growth for the nonlocal term and critical nonlinearity. Such behaviour causes new difficulties to treat our study from an exclusively variational point of view, besides those already known for the local operators. Some tools we used in this paper are the mountain-pass and Ekeland’s Theorems and the Lions’ Concentration Compactness Principle.

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Acknowledgements

The first author was supported by Programa de Incentivo à Pós–Graduação e Pesquisa (PROPESQ 1.01.02.05-1-337) Edital 2015, UEPB. The authors would like to express your gratitude to the anonymous referee for his(her) carefully reading of the manuscript with valuable comments and suggestions.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Estadual da Paraíba, CEP: 58429-500, Campina Grande, PB, Brazil

    Francisco S. B. Albuquerque

  2. Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP: 58429-900, Campina Grande, PB, Brazil

    Marcelo C. Ferreira

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  1. Francisco S. B. Albuquerque
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  2. Marcelo C. Ferreira
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Correspondence to Marcelo C. Ferreira.

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Albuquerque, F.S.B., Ferreira, M.C. A Nonhomogeneous and Critical Kirchhoff–Schrödinger Type Equation in \(\mathbb R^4\) Involving Vanishing Potentials. Mediterr. J. Math. 18, 189 (2021). https://doi.org/10.1007/s00009-021-01829-y

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