Opuscula Math. 41, no. 1 (2021), 5-23

Opuscula Mathematica

Some existence results for a nonlocal non-isotropic problem

Rachid Bentifour
Sofiane El-Hadi Miri

Abstract. In this paper we deal with the following problem \[\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}\] where \(\Omega\) is a bounded regular domain in \(\mathbb{R}^{N}\). We will assume without loss of generality that \(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^{m}(\Omega)\), \(1\lt q\lt \overline{p}^{\ast}\), \(a\gt 0\), \(b\gt 0\) and \(0\lt\gamma \lt 1.\)

Keywords: anisotropic operator, integro-differential problem, variational methods.

Mathematics Subject Classification: 35A15, 35B09, 35E15, 35J20.

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  • Rachid Bentifour
  • ORCID iD https://orcid.org/0000-0003-3177-663X
  • Université de Tlemcen, Dépatement GEE, Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, BP 119 Tlemcen, 13000, Algeria
  • Sofiane El-Hadi Miri (corresponding author)
  • ORCID iD https://orcid.org/0000-0001-6572-4366
  • Université de Tlemcen, Dépatement GEE, Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, BP 119 Tlemcen, 13000, Algeria
  • Communicated by Vicentiu D. Radulescu.
  • Received: 2020-06-01.
  • Revised: 2020-11-23.
  • Accepted: 2020-11-24.
  • Published online: 2021-02-08.
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Cite this article as:
Rachid Bentifour, Sofiane El-Hadi Miri, Some existence results for a nonlocal non-isotropic problem, Opuscula Math. 41, no. 1 (2021), 5-23, https://doi.org/10.7494/OpMath.2021.41.1.5

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