Opuscula Math. 44, no. 6 (2024), 789-814
https://doi.org/10.7494/OpMath.2024.44.6.789

 
Opuscula Mathematica

On a nonlocal p(x)-Laplacian Dirichlet problem involving several critical Sobolev-Hardy exponents

Augusto César dos Reis Costa
Ronaldo Lopes da Silva

Abstract. The aim of this work is to present a result of multiplicity of solutions, in generalized Sobolev spaces, for a non-local elliptic problem with \(p(x)\)-Laplace operator containing \(k\) distinct critical Sobolev-Hardy exponents combined with singularity points \[ \begin{cases} M(\psi(u)) (- \Delta_{p(x)} u + |u|^{p(x)-2} u) = \sum_{i=1}^{k} h_i(x) \dfrac{|u|^{p^*_{s_i}(x)-2} u}{|x|^{s_i(x)}} + f(x,u) &\text{in }\Omega, \\ u=0 &\text{on }\partial \Omega, \end{cases} \] where \(\Omega\subset \mathbb{R}^N\) is a bounded domain with \(0 \in \Omega\) and \(1 \lt p^- \leq p(x) \leq p^+ \lt N\). The real function \(M\) is bounded in \([0, +\infty)\) and the functions \(h_i\) \((i=1, \ldots, k)\) and \(f\) are functions whose properties will be given later. To obtain the result we use the Lions' concentration-compactness principle for critical Sobolev-Hardy exponent in the space \(W^{1,p(x)}_{0}(\Omega)\) due to Yu, Fu and Li, and the Fountain Theorem.

Keywords: generalized Lebesgue-Sobolev spaces, \(p(x)\)-Laplacian nonlocal operator, Sobolev-Hardy critical exponents, concentration-compactness principle for critical Sobolev-Hardy exponent, fountain theorem.

Mathematics Subject Classification: 35J60, 58E05.

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  • Augusto César dos Reis Costa
  • ORCID iD https://orcid.org/0000-0002-9798-5357
  • Universidade Federal do Pará (UFPA), Instituto de Ciências Exatas e Naturais, R. Augusto Corrêa, 01 - Guamá, CEP 66075-110, Belém, PA, Brasil
  • Ronaldo Lopes da Silva
  • Universidade Federal do Pará (UFPA), Instituto de Ciências Exatas e Naturais, R. Augusto Corrêa, 01 - Guamá, CEP 66075-110, Belém, PA, Brasil
  • Communicated by Giovany Figueiredo.
  • Received: 2024-04-08.
  • Revised: 2024-06-27.
  • Accepted: 2024-07-02.
  • Published online: 2024-10-11.
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Cite this article as:
Augusto César dos Reis Costa, Ronaldo Lopes da Silva, On a nonlocal p(x)-Laplacian Dirichlet problem involving several critical Sobolev-Hardy exponents, Opuscula Math. 44, no. 6 (2024), 789-814, https://doi.org/10.7494/OpMath.2024.44.6.789

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